Derived Functors

Daniel Murfet

October 5, 2006

In this note we give an exposition of some basic topics in homological algebra. Most of thismaterial can be found in either [3] or [2], but for some topics the best reference is still [1].

Contents

1 Definitions 1

2 Projective Resolutions 6

3 Left Derived Functors 8

4 Injective Resolutions 10

5 Right Derived Functors 12

6 The Long Exact (Co)Homology Sequence 14

7 Two Long Exact Sequences 187.1 Left Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Right Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Dimension Shifting 278.1 Acyclic Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Change of Base 33

10 Homology and Colimits 37

11 Cohomology and Limits 38

12 Delta Functors 39

1 Definitions

Throughout we work in an abelian category A. We assume a strong axiom of choice that allowsus to associate to every morphism f : A −→ B a canonical kernel Ker(f) −→ A, cokernelB −→ Coker(f), and image Im(f) −→ B. Similarly we pick a canonical zero object 0, and acanonical biproduct for any finite nonempty family of objects. We say A is a category of modulesif it is ModR or RMod for some ring R (in particular Ab is a category of modules), and fora category of modules we choose the obvious canonical structures, with the subtle exception ofchoosing the identity 1A : A −→ A to be the cokernel of any morphism 0 −→ A from a zero object.For a subobject u : A −→ B we write B −→ B/A for the cokernel. Given subobjects u : A −→ Band v : C −→ B we write u ≤ v to mean that u factors through v. The factorisation A −→ C ismonic, so we can talk about the quotient C/A.

1

Definition 1. A chain complex C in A is a collection of objects {Cn}n∈Z together with morphisms∂n : Cn −→ Cn−1 satisfying ∂n∂n+1 = 0. We write a chain complex as a descending chain

· · · // Cn+1∂n+1 // Cn

∂n // Cn−1// · · ·

The morphisms ∂n are called the differential (or boundary operators). A morphism of chaincomplexes ψ : C −→ D is a collection ψ of morphisms {ψn : Cn −→ Dn}n∈Z satisfying ψn∂n+1 =∂n+1ψn+1 for all n ∈ Z. That is, they fit into a commutative diagram

· · · // Cn+1

ψn+1

��

∂n+1 // Cn

ψn

��

∂n // Cn−1

ψn−1

��

// · · ·

· · · // Dn+1∂n+1

// Dn∂n

// Dn−1// · · ·

Composition is defined by (ψϕ)n = ψnϕn and addition by (ψ+ϕ)n = ψn +ϕn, which defines thepreadditive category ChA of chain complexes in A. Since ∂n∂n+1 = 0 we have Im∂n+1 ≤ Ker∂nfor all n ∈ Z. The object Hn(C) = Ker∂n/Im∂n+1 is called the n-th homology object of the chaincomplex C.

Definition 2. A cochain complex C in A is a collection of objects {Cn}n∈Z together with mor-phisms ∂n : Cn −→ Cn+1 satisfying ∂n+1∂n = 0. We write a cochain complex as an ascendingchain

· · · // Cn−1 ∂n−1// Cn

∂n// Cn+1 // · · ·

The morphisms ∂n are called the coboundary operators. A morphism of cochain complexes ψ :C −→ D is a collection ψ of morphisms {ψn : Cn −→ Dn}n∈Z satisfying ψn+1∂n = ∂nψn for alln ∈ Z. That is, they fit into a commutative diagram

· · · // Cn−1

ψn−1

��

∂n−1// Cn

ψn

��

∂n// Cn+1

ψn+1

��

// · · ·

· · · // Dn−1

∂n−1// Dn

∂n// Dn+1 // · · ·

(1)

Composition is defined by (ψϕ)n = ψnϕn and addition by (ψ+ϕ)n = ψn +ϕn, which defines thepreadditive category coChA of cochain complexes in A. Since ∂n+1∂n = 0 we have Im∂n−1 ≤Ker∂n for all n ∈ Z. The object Hn(C) = Ker∂n/Im∂n−1 is called the n-th cohomology objectof the cochain complex C.

Lemma 1. There is an isomorphism of categories ChA ∼= coChA.

Proof. Given a chain complex C = {Cn, ∂n}n∈Z define the cochain complex F (C) = {Dn, δn}n∈Zby Dn = D−n and δn = ∂−n. Given a morphism ψ : C −→ C ′ define F (ψ) : F (C) −→ F (C ′) byF (ψ)n = ψ−n. This is clearly an isomorphism.

Lemma 2. Let ϕ : C −→ D be a morphism of cochain complexes. Then

(i) ϕ is an isomorphism iff. ϕn is an isomorphism for all n ∈ Z;

(ii) ϕ is an epimorphism iff. ϕn is an epimorphism for all n ∈ Z;

(iii) ϕ is a monomorphism iff. ϕn is a monomorphism for all n ∈ Z.

The same result is true for chain complexes.

2

Proof. (i) is easily checked. Suppose g : Dn −→ E is given with gϕn = 0. We have to show thatg = 0. We can fit g into a morphism of cochains with entries as depicted in the following diagram

· · · // Cn−2

��

// Cn−1

��

// Cn

ϕn

��

// Cn+1

��

// · · ·

· · · // Dn−2

��

// Dn−1 ∂n−1//

g∂n−1

��

Dn

g

��

// Dn+1

��

// · · ·

· · · // 0 // E1

// E // 0 // · · ·

The composite of these two cochain morphisms is zero, so since ϕ is an epimorphism it followsthat g = 0, as required. A similar construction proves (iii), and it is clear how to translate theseresults into statements about chain complexes.

The categories ChA and coChA are abelian. First we give the definitions for coChA.

Zero The 0 cochain is · · · −→ 0 −→ 0 −→ 0 −→ · · · .

Finite products Given cochains C1, . . . , Cr we define

(C1 ⊕ · · · ⊕ Cr)n = Cn1 ⊕ · · · ⊕ Cnr

∂n = ∂n1 ⊕ · · · ⊕ ∂nr

It is not difficult to see that this is a cochain complex. The injection ui : Ci −→ C1⊕· · ·⊕Cris pointwise the injection Cni −→ Cn1 ⊕ · · · ⊕Cnr , and the projection pi is also pointwise theprojection Cn1 ⊕ · · · ⊕ Cnr −→ Cni . It is easy to check that piuj = δij and

∑ukpk = 1 so

that these morphisms are indeed a biproduct.

Kernels and Cokernels Let ϕ : C −→ D be a morphism of cochain complexes. For each n ∈ Zthe commutative diagram (1) induces morphisms Ker(ϕn) −→ Ker(ϕn+1) for all n whichare unique making the following diagram commute:

· · · // Ker(ϕn)

��

// Ker(ϕn+1)

��

// · · ·

· · · // Cn //

��

Cn+1

��

// · · ·

· · · // Dn // Dn+1 // · · ·

This cochain is a kernel for ϕ, where given any ψ : E −→ C with ϕψ = 0 the uniquefactorisation through Kerϕ −→ C is pointwise the unique factorisation of ψn throughKer(ϕn) −→ Cn. A dual situation holds for cokernels.

Normal and Conormal Given our description of the kernel and cokernel, it is clear that sinceA is normal and conormal that every monomorphism in coChA is the kernel of its cokernel,and every epimorphism is the cokernel of its kernel.

Epi-Mono Factorisations Let ϕ : C −→ D be a morphism of cochains. The morphismsIm(ϕn) −→ Dn are kernels for Dn −→ Coker(ϕn) so we induce morphisms Im(ϕn) −→Im(ϕn+1) which form a cochain Im(ϕ). The factorisations Cn −→ Im(ϕn) give a morphismC −→ Im(ϕ) which is a pointwise epimorphism and therefore an epimorphism of cochains.Similarly Im(ϕ) −→ D is a pointwise monomorphism and therefore a monomorphism ofcochains, so we have the desired factorisation. Note that Im(ϕ) −→ D is the image of ϕ incoChA.

3

This completes the proof that coChA is abelian. The obvious translation of these structuresto chains gives the definition of zero, biproducts, kernels, cokernels and images, which showsthat ChA is an abelian category. By picking a canonical zero object, canonical finite nonemptyproducts and canonical kernels, cokernels and images for the category A we get canonical choicesfor all these structures in coChA and ChA. Note that a sequence

Cϕ // // D

ψ // E

of chains is exact in ChA if and only if for all n ∈ Z the following sequence is exact in A

Cnϕn //// Dn

ψn // En

The same is true of cochains.

Lemma 3. Let D be a diagram in coChA and suppose we have a cocone {αi : Di −→ X}i∈I onthis diagram with the property that {αni : Dn

i −→ Xn} is a colimit for the diagram Dn in A forevery n ∈ Z. Then the αi are a colimit for D in coChA.

Lemma 4. Let D be a diagram in coChA and suppose we have a cone {αi : X −→ Di}i∈I onthis diagram with the property that {αni : Xn −→ Dn

i } is a limit for the diagram Dn in A forevery n ∈ Z. Then the αi are a limit for D in coChA.

Suppose we have a commutative diagram in A

Af //

��

B

��C g

// D

Then there are unique morphisms Ker(f) −→ Ker(g), Coker(f) −→ Coker(g) and Im(f) −→Im(g) making the following diagrams commute

Ker(f) //

��

A

��Ker(g) // C

B //

��

Coker(f)

��D // Coker(g)

Im(f) //

��

B

��Im(g) // D

Let ϕ : C −→ D be a morphism of cochains in coChA. For any n ∈ Z we induce a morphismHn(C) −→ Hn(D) which is unique making the following diagram commute

Im(∂n−1) //

$$JJJJJJJJJ

��

Ker(∂n)

��

zzvvvvvvvvv// Hn(C)

Hn(ϕ)

��

Cn−1

ϕn−1

��

99ssssssssss// Cn

ϕn

��

// Cn+1

ϕn+1

��Dn−1 //

%%KKKKKKKKKK Dn // Dn+1

Im(∂n−1) //

::uuuuuuuuuKer(∂n)

ddHHHHHHHHH// Hn(D)

(2)

Hence Hn(−) defines a covariant additive functor coChA −→ A. A similar argument shows thatHn(−) defines a covariant additive functor ChA −→ A.

4

Definition 3. A homotopy Σ : ϕ −→ ψ between two chain morphisms ϕ,ψ : C −→ D is acollection of morphisms Σn : Cn −→ Dn+1 such that ψ − ϕ = ∂Σ + Σ∂. That is,

ψn − ϕn = ∂n+1Σn + Σn−1∂n ∀n ∈ Z

as in the diagram

· · · // Cn+1∂n+1 //

��

Cn

ψn

��

ϕn

��

Σn

}}{{{{

{{{{

{{{{

{{{{

{

∂n // Cn−1

��

//

Σn+1

}}{{{{

{{{{

{{{{

{{{{

{· · ·

· · · // Dn+1∂n+1

// Dn∂n

// Dn−1// · · ·

We say that ϕ,ψ are homotopic, and write ϕ ' ψ if there exists a homotopy Σ : ϕ −→ ψ. IfΣ is such a homotopy, then the morphisms −Σn define a homotopy −Σ : ψ −→ ϕ, so there isa bijection between homotopies ϕ −→ ψ and homotopies ψ −→ ϕ. We denote the set of allhomotopies ϕ −→ ψ by Hom(ϕ,ψ).

Lemma 5. The homotopy relation ' is an equivalence relation, which is stable under composition.That is, if ϕ,ψ : C −→ D are chain morphisms with ϕ ' ψ then for any chain morphismsγ : B −→ C and τ : D −→ E we have τϕ ' τψ and ϕγ ' ψγ.

Proof. The relation is clearly reflective and symmetric. To check transitivity, let ψ−ϕ = ∂Σ+Σ∂and ξ − ψ = ∂T + T∂. An easy calculation shows that ξ − ϕ = ∂(Σ + T ) + (Σ + T )∂. For theresults about composition, let us assume that ψ − ϕ = ∂Σ + Σ∂. Then the morphisms τn+1Σngive rise to a homotopy τψ ' τϕ and the morphisms Σnγn give rise to a homotopy ψγ ' ϕγ.

Proposition 6. Let ϕ,ψ : C −→ D be homotopic chain morphisms. Then Hn(ϕ) = Hn(ψ) forevery n ∈ Z.

Proof. It suffices to show that if ϕ ' 0 then Hn(ϕ) = 0 for all n ∈ Z. Let Σ : ϕ −→ 0 be ahomotopy, so ϕn = ∂n+1Σn + Σn−1∂n for all n. Let n ∈ Z be fixed, and let k : Ker∂n −→ Cn bethe kernel of ∂n. Then

ϕnk = (∂n+1Σn + Σn−1∂n)k = ∂n+1Σnk

So ϕnk factors through Im∂n+1 −→ Dn, from which we conclude that in the chain analogue of (2)there is a morphism Ker∂n −→ Im∂n+1 making the whole diagram commute. So the compositeKer∂n −→ Ker∂n −→ Hn(D) must be zero, and therefore Hn(ϕ) = 0, as required.

Lemma 7. Let F : A −→ B be an additive functor between abelian categories. Then F induces anadditive functor ChA −→ ChB (which we denote by the same symbol) defined by F (C)n = F (Cn)and F (ϕ)n = F (ϕn). This functor has the following properties

(a) If Σ : ϕ −→ ψ is a homotopy then so is F (Σ) : F (ϕ) −→ F (ψ).

(b) If ϕ ' ψ then Hn(F (ϕ)) = Hn(F (ψ)) for all n ∈ Z.

(c) If F is exact, so is the induced functor ChA −→ ChB.

Now for the cochain version of homotopy:

Definition 4. A homotopy Σ : ϕ −→ ψ between two cochain morphisms ϕ,ψ : C −→ D is acollection of morphisms Σn : Cn −→ Dn−1 such that ψ − ϕ = ∂Σ + Σ∂. That is,

ψn − ϕn = ∂n−1Σn + Σn+1∂n ∀n ∈ Z

5

as in the diagram

· · · // Cn−1 ∂n−1//

��

Cn

ψn

ϕn

��

Σn

}}{{{{

{{{{

{{{{

{{{{

{∂n

// Cn+1

��

//

Σn+1

}}{{{{

{{{{

{{{{

{{{{

{· · ·

· · · // Dn−1

∂n−1// Dn

∂n// Dn+1 // · · ·

We say that ϕ,ψ are homotopic, and write ϕ ' ψ if there exists a homotopy Σ : ϕ −→ ψ. IfΣ is such a homotopy, then the morphisms −Σn define a homotopy −Σ : ψ −→ ϕ, so there isa bijection between homotopies ϕ −→ ψ and homotopies ψ −→ ϕ. We denote the set of allhomotopies ϕ −→ ψ by Hom(ϕ,ψ).

The following results are proved just as in the chain case:

Lemma 8. The homotopy relation ' is an equivalence relation, which is stable under composition.That is, if ϕ,ψ : C −→ D are cochain morphisms with ϕ ' ψ then for any cochain morphismsγ : B −→ C and τ : D −→ E we have τϕ ' τψ and ϕγ ' ψγ.

Proposition 9. Let ϕ,ψ : C −→ D be homotopic cochain morphisms. Then Hn(ϕ) = Hn(ψ) forevery n ∈ Z.

Lemma 10. Let F : A −→ B be an additive functor between abelian categories. Then F inducesan additive functor coChA −→ coChB (which we denote by the same symbol) defined by F (C)n =F (Cn) and F (ϕ)n = F (ϕn). This functor has the following properties

(a) If Σ : ϕ −→ ψ is a homotopy then so is F (Σ) : F (ϕ) −→ F (ψ).

(b) If ϕ ' ψ then Hn(F (ϕ)) = Hn(F (ψ)) for all n ∈ Z.

(c) If F is exact, so is the induced functor coChA −→ coChB.

2 Projective Resolutions

Throughout we work with an abelian category A and chain complexes over A.

Definition 5. A chain complex C is positive if Cn = 0 for all n < 0. In other words the complexlooks like

· · · −→ Cn −→ Cn−1 −→ · · · −→ C1 −→ C0 −→ 0

A positive chain complex C is projective if Cn is projective for all n ≥ 0. It is called acyclic ifHn(C) = 0 for n ≥ 1. A positive chain complex C is acyclic if and only if the following sequenceis exact

· · · −→ Cn −→ Cn−1 −→ · · · −→ C1 −→ C0 −→ H0(C) −→ 0

A projective resolution of an object A is a projective acyclic chain complex P together with amorphism P0 −→ A making the following sequence exact

· · · −→ Pn −→ Pn−1 −→ · · · −→ P1 −→ P0 −→ A −→ 0 (3)

Equivalently, it is an exact sequence (3) with all the Pi projective.

Lemma 11. The category Ch+A of positive chain complexes is an abelian subcategory of ChA.

Proof. The canonical zero, products, kernels and cokernels for a morphism of positive chain com-plexes are themselves positive, so it is clear that Ch+A is an abelian subcategory. In particularthis means that a sequence C ′ −→ C −→ C ′′ in Ch+A is exact iff. it is exact in ChA, so iff.C ′n −→ Cn −→ C ′′n is exact in A for all n ≥ 0.

6

If C is a positive chain complex then the construction of homology gives an epimorphismC0 −→ H0(C) which is the cokernel of Im(∂1) and therefore of C1 −→ C0. If ϕ : C −→ D isa morphism of positive chain complexes then H0(ϕ) : H0(C) −→ H0(D) is just the morphisminduced between the cokernels by the following commutative diagram

C1//

ϕ1

��

C0//

ϕ0

��

H0(C) //

H0(ϕ)

��

0

D1∂1

// D0// H0(D) // 0

Theorem 12. Let C,D be positive chain complexes with C projective and D acyclic. Then themap ϕ 7→ H0(ϕ) gives a bijection between homotopy classes of chain morphisms C −→ D andmorphisms H0(C) −→ H0(D).

Proof. First we show that given any morphism ϕ : H0(C) −→ H0(D) there is a chain morphisminducing ϕ on homology. The chain morphism ψ : C −→ D is defined recursively. Consider thefollowing diagram, whose bottom row is exact

C1//

ψ1

��

C0//

ψ0

��

H0(C) //

ϕ

��

0

D1∂1

// D0// H0(D) // 0

Since C0 is projective we can induce ψ0 : C0 −→ D0 making the right hand square commute. Sincethe bottom row is exact at D0 and ∂ψ0∂ = 0, we see that ψ0∂ factors through Im(∂1) −→ D0.Since C1 is projective and D1 −→ Im(∂1) is an epimorphism, we induce ψ1 making the abovediagram commute. For n ≥ 2 we use the exactness of D at n − 1 and the same argument toproduce ψn. It is clear that ψ is a chain morphism which induces ϕ on the cokernels.

Now suppose ϕ,ψ are two chain morphisms inducing the same morphism H0(C) −→ H0(D).Recursively we define a homotopy Σ : ψ −→ ϕ. First consider the diagram

C1//

ψ1

ϕ1

��

C0

Σ0xx

//

ψ0

ϕ0

��

H0(C) //

��

0

D1∂1

// D0f

// H0(D) // 0

Since ϕ0, ψ0 induce the same morphism on the cokernel, we must have ϕ0 − ψ0 factoring throughKerf = Im∂1. Since C0 is projective and D1 −→ Im(∂1) is an epimorphism, there is Σ0 : C0 −→D1 with ∂1Σ0 = ϕ0 − ψ0. Suppose n ≥ 1 and that Σ0, . . . ,Σn−1 have been defined in such a waythat ϕr − ψr = ∂Σr + Σr−1∂ for r ≤ n− 1 (with Σ−1∂ being understood as 0), as in the diagram

Cn+1//

ψn+1

ϕn+1

��

Cn

Σnww

//

ψn

ϕn

��

Cn−1

Σn−1

wwoooooooooooooψn−1

ϕn−1

��Dn+1

// Dn// Dn−1

We have∂(ϕn − ψn − Σn−1∂) = (ϕn−1 − ψn−1 − ∂Σn−1)∂ = Σn−2∂∂ = 0

Hence ϕn − ψn − Σn−1∂ factors through Im(∂n+1 : Dn+1 −→ Dn), and using the fact that Cn isprojective we obtain Σn : Cn −→ Dn+1 with the required property. Hence ψ ' ϕ, as claimed. Ofcourse if ψ ' ϕ then H0(ϕ) = H0(ψ), which completes the proof.

So this one situation where equality on homology at 0 always arises from a homotopy. Animportant special case is the following.

7

Corollary 13. Suppose we are given projective resolutions of objects A,B in A

P : · · · −→ P1 −→ P0 −→ A −→ 0Q : · · · −→ Q1 −→ Q0 −→ B −→ 0

Then there is a bijection between homotopy classes of chain morphisms P −→ Q and morphismsA −→ B. Given a chain morphism ϕ : P −→ Q the corresponding morphism α : A −→ B isunique making the following diagram commute

P0

ϕ0

��

// A

α

��Q0

// B

3 Left Derived Functors

Let A be an abelian category. We say that A has enough projectives if for every object A there isa projective object P0 and an epimorphism P0 −→ A. If A has enough projectives then it is clearthat we can construct a projective resolution for any object A by taking the kernel K −→ P0,finding another epimorphism P1 −→ K with P1 projective, and repeating the process. Throughoutthis section A is an abelian category with enough projectives, and B is any abelian category.

Let T : A −→ B be an additive covariant functor. Suppose we have a projective resolution ofA

P : · · · −→ P1 −→ P0 −→ A −→ 0

This gives rise to a chain complex of objects of B

TP : · · · −→ T (Pn) −→ · · · −→ T (P1) −→ T (P0) −→ 0

We define LPnT (A) = Hn(TP ) for n ≥ 0. Let B be another object with projective resolution Qand let α : A −→ B be a morphism. By Corollary 13 there is a chain morphism ϕ : P −→ Qmaking a commutative diagram with exact rows

· · · // P1//

ϕ1

��

P0//

ϕ0

��

A //

α

��

0

· · · // Q1// Q0

// B // 0

For n ≥ 0 there is a morphism Hn(T (ϕ)) : LPnT (A) −→ LQn T (B), which depends only on α andnot on the particular ϕ we use in the construction (since all candidate ϕ are homotopic).

Let P be an assignment of a projective resolution to every object of A (we need a strong axiomof choice to show that such an assignment exists) and for fixed n ≥ 0 let LnT (A) denote the objectcalculated with the projective resolution chosen by P. Let LnT (α) : LnT (A) −→ LnT (B) denotethe morphism induced by α : A −→ B. Then it is easy to check that LnT : A −→ B is a covariantadditive functor. If necessary we denote this functor by LPn T to indicate its dependence on P.

Proposition 14. Let T : A −→ B be an additive covariant functor. If P and Q are two as-signments of projective resolutions to objects of A, then for n ≥ 0 there is a canonical naturalequivalence LPn T ∼= LQn T .

Proof. For an object A the morphism 1A : A −→ A gives rise to chain morphisms ξ : P −→ Qand η : Q → P of the corresponding projective resolutions. Moreover, ξη and ηξ induce theidentity on A and so by Corollary 13 they must be homotopic to the identity chain morphism.Applying T and passing to homology we obtain an isomorphism LPn T (A) ∼= LQn T (A) which iseasily checked to be natural in A. Since the isomorphism depends only on the homotopy class ofξ, η this isomorphism is canonical.

8

Definition 6. Let T : A −→ B be a covariant additive functor. For n ≥ 0 the covariant additivefunctor LnT : A −→ B is called the n-th left derived functor of T . This functor depends onthe projective resolutions chosen, but is independent of these choices up to canonical naturalequivalence.

Proposition 15. If T : A −→ B is right exact, then L0T and T are canonically naturallyequivalent. If T is exact then LnT ∼= 0 for n 6= 0.

Proof. Let P be a projective resolution of A. Then P1 −→ P0 −→ A −→ 0 is exact, so TP1 −→TP0 −→ TA −→ 0 is exact. Hence H0(TP ) ∼= TA. This isomorphism is readily seen to be naturalin A. It is clear that if T is exact all the higher left derived functors are zero.

Proposition 16. For a projective object P , LnT (P ) = 0 for n ≥ 1 and L0T (P ) ∼= T (P ).

Proof. This follows immediately from the fact that the chain complex P with Pi = 0 except forP0 = P is a projective resolution of P .

The next result shows how to calculate the values of left derived functors of functors withvalues in a category of modules from a “truncated” projective resolution.

Proposition 17. Suppose we have an exact sequence in A

0 // Kµ // Pn // Pn−1

// · · · // P0// A // 0 (4)

where P0, . . . , Pn are all projective and n ≥ 0. If B is a category of modules and T : A −→ B isright exact then there is an exact sequence

0 // Ln+1T (A) // T (K)T (µ) // T (Pn)

Proof. Find a projective Pn+1 and an epimorphism e : Pn+1 −→ K and in the usual way continuethis process to produce a projective resolution P whose differential ∂n+1 : Pn+1 −→ Pn is µe.Since T is right exact, the following diagram is commutative with exact rows

T (Pn+2)

��

T (∂n+2)// T (Pn+1)

T (∂n+1)

��

T (e) // T (K)

T (µ)

��

// 0

0 // 0 // T (Pn) +3 T (Pn)

The Snake Lemma provides us with an exact sequence

T (Pn+2) −→ KerT (∂n+1) −→ KerT (µ) −→ 0

But of course Ln+1T (A) = KerT (∂n+1)/ImT (∂n+2) so the modules KerT (µ) and Ln+1T (A)are isomorphic. The morphism Ln+1T (A) −→ T (K) fitting into the exact sequence is actuallyx+ ImT (∂n+2) 7→ T (e)(x).

This idea can be pushed further. Suppose for some fixed n ≥ 0 we have an exact sequence ofthe form (4) for every object A of A, which we can extend to a projective resolution. Define theadditive functor `n+1T : A −→ B as follows. The module `n+1T (A) is KerT (µ). A morphismα : A −→ B can be lifted to a chain morphism which induces α′ : K −→ M in the followingdiagram

0 // K

α′

��

µ // Pn

��

// Pn−1//

��

· · · // P0

��

// A

α

��

// 0

0 // M τ// Qn // Qn−1

// · · · // Q0// B // 0

9

Let `n+1T (α) : KerT (µ) −→ KerT (τ) be x 7→ T (α′)(x). As we will see in a moment, this isindependent of the chain morphism lifting α and defines an additive functor `n+1T .

If Ln+1T is defined using projective resolutions obtained from these exact sequences, then itis not hard to see that there is a commutative diagram

KerT (µ)

`n+1T (α)

��

// T (K)

��

T (µ) // T (Pn)

��

Ln+1T (A)

bj MMMMMMMMMM

MMMMMMMMMM

99ssssssssss

��

KerT (τ) // T (M)T (τ)

// T (Qn)

Ln+1T (B)

bj MMMMMMMMMM

MMMMMMMMMM

99ssssssssss

which shows that `n+1T is well-defined and is naturally equivalent to Ln+1T . So providedT : A −→ B is a right exact functor into a category of modules we can calculate the left de-rived functors L1T,L2T, . . . using finite exact sequences instead of projective resolutions.

Let A be an abelian category with enough injectives and T : A −→ B a contravariant additivefunctor. Then Aop is an abelian category with enough projectives, so for n ≥ 0 we can define theleft derived functor LnT of T ∗, which is a contravariant additive functor LnT : A −→ B. Let Ibe an assignment of injective resolutions to objects of A. Then this is an assignment of projectiveresolutions to objects ofAop and we calculate LnT as follows: let I : 0 −→ A −→ I0 −→ I1 −→ · · ·be the injective resolution of A, and consider the chain complex of objects of B

TI : · · · −→ T (I2) −→ T (I1) −→ T (I0) −→ 0

Then LnT (A) is the homology object Hn(TP ). Given α : A −→ A′ let ϕ : I −→ I ′ be the chainmorphism inducing α. Then T (ϕ) : TI ′ −→ TI is a chain morphism and LnT (α) : LnT (A′) −→LnT (A) is HnT (ϕ). We call LnT the n-th left derived functors of T .

4 Injective Resolutions

Throughout we work with an abelian category A and cochain complexes over A.

Definition 7. A cochain complex C is positive if Cn = 0 for all n < 0. In other words thecomplex looks like

0 −→ C0 −→ C1 −→ · · · −→ Cn −→ · · ·

A positive cochain complex C is injective if Cn is injective for all n ≥ 0. It is called acyclic ifHn(C) = 0 for n ≥ 1. A positive cochain complex C is acyclic if and only if the following sequenceis exact

0 −→ H0(C) −→ C0 −→ C1 −→ · · · −→ Cn −→ · · ·

An injective resolution of an object A is an injective acyclic cochain complex I together with amorphism A −→ I0 making the following sequence exact

0 −→ A −→ I0 −→ I1 −→ · · · −→ In −→ · · · (5)

Equivalently, it is an exact sequence (5) with all the Ii injective.

Lemma 18. The category coCh+A of positive cochain complexes is an abelian subcategory ofcoChA.

10

Proof. As before. Once again, we note that a sequence C ′ −→ C −→ C ′′ in coCh+A is exact iff.C ′n −→ Cn −→ C ′′n is exact in A for n ≥ 0.

If C is a positive cochain complex then the construction of cohomology gives a monomorphismH0(C) −→ C0 which is kernel of C0 −→ C1. If ϕ : C −→ D is a morphism of positive cochaincomplexes then H0(ϕ) : H0(C) −→ H0(D) is just the morphism induced between the kernels bythe following commutative diagram

0 // H0(C) //

H0(ϕ)

��

C0 //

ϕ0

��

C1

ϕ1

��0 // H0(D) // D0 // D1

Theorem 19. Let C,D be positive cochain complexes with C acyclic and D injective. Then themap ϕ 7→ H0(ϕ) gives a bijection between homotopy classes of cochain morphisms C −→ D andmorphisms H0(C) −→ H0(D).

Proof. One could argue that since the dual of an abelian category is abelian, this follows fromTheorem 12 by duality (note that now the morphisms go from the acyclic to the injective cochain,whereas before they went from the projective to the acyclic chain). Or one can just copy the proofof Theorem 12. First we show that given any morphism ϕ : H0(C) −→ H0(D) there is a chainmorphism inducing ϕ on cohomology. The chain morphism ψ : C −→ D is defined recursively.Consider the following diagram, whose top row is exact:

0 // H0(C) //

ϕ

��

C0 //

ψ0

��

C1

ψ1

��0 // H0(D) // D0 // D1

The fact that D0 is injective means that we can find ψ0 making the diagram commute. Since thetop row is exact, C0 −→ Im(∂0) is the cokernel of H0(C) −→ C0. Since ∂ψ0∂ = 0 we see thatC0 −→ D0 −→ Im(∂0) factors through C0 −→ Im(∂0) and using the fact that D1 is injective weget ψ1 making the diagram commute. Proceeding in this way we construct the cochain morphismψ, which clearly induces ϕ on cohomology.

Now suppose ϕ,ψ are two chain morphisms inducing the same morphism H0(C) −→ H0(D).Recursively we define a homotopy Σ : ψ −→ ϕ. First consider the diagram

0 // H0(C) //

��

C0 ∂0//

ϕ0

ψ0

��

C1

Σ1

xxϕ1

ψ1

��0 // H0(D) // D0 // D1

Since ϕ0 − ψ0 gives zero on composition with H0(C) −→ C0 we see that ϕ0 − ψ0 factors throughC0 −→ Im(∂0). Since D0 is injective we obtain Σ1 with ϕ0 − ψ0 = Σ1∂0. Suppose n ≥ 1 andthat Σ0, . . . ,Σn−1 have been defined in such a way that ϕr − ψr = Σr+1∂ + ∂Σr for all r ≤ n− 1(with Σ0 being understood as 0). Consider the diagram

Cn−2 //

ϕn−2

ψn−2

��

Cn−1

Σn−1vvnnnnnnnnnnnnn

//

ϕn−1

ψn−1

��

Cn

Σn

wwϕn

��ψn

��Dn−2 // Dn−1 // Dn

We have

(−∂Σn−1 + ϕn−1 − ψn−1)∂ = ∂(ϕn−2 − ψn−2 − Σn−1∂) = ∂∂Σn−2 = 0

Once again we factor through the morphism Cn−1 −→ Im(∂n−1) and use injectivity of Dn−1 toobtain Σn with Σn∂ + ∂Σn−1 = ϕn−1 − ψn−1. Hence ψ ' ϕ, as claimed.

11

Corollary 20. Suppose we are given injective resolutions of objects A,B in A

I : 0 −→ A −→ I0 −→ I1 −→ · · ·J : 0 −→ B −→ J0 −→ J1 −→ · · ·

Then there is a bijection between homotopy classes of cochain morphisms I −→ J and morphismsA −→ B. Given a cochain morphism ϕ : I −→ J the corresponding morphism α : A −→ B isunique making the following diagram commute

A

α

��

// I0

ϕ0

��B // J0

5 Right Derived Functors

Let A be an abelian category. We say that A has enough injectives if for every object A there is aninjective object I0 and a monomorphism A −→ I0. If A has enough injectives then it is clear thatwe can construct an injective resolution for any object A by taking the cokernel I0 −→ C, findinganother monomorphism C −→ I1 with I1 injective, and repeating the process. Throughout thissection A is an abelian category with enough injectives, and B is any abelian category.

Let T : A −→ B be an additive covariant functor. Suppose we have an injective resolution ofA

I : 0 −→ A −→ I0 −→ I1 −→ · · ·

This gives rise to a cochain complex of objects of B

TI : 0 −→ T (I0) −→ T (I1) −→ · · · −→ T (In) −→ · · ·

We define RnI T (A) = Hn(TI) for n ≥ 0. Let B be another object with injective resolution J andlet α : A −→ B be a morphism. By Corollary 20 there is a chain morphism ϕ : I −→ J making acommutative diagram with exact rows

0 // A //

α

��

I0 //

ϕ0

��

I1 //

ϕ1

��

· · ·

0 // B // J0 // J1 // · · ·

For n ≥ 0 there is a morphism Hn(T (ϕ)) : RnI T (A) −→ RnJT (B), which depends only on α andnot on the particular ϕ we use in the construction (since all candidate ϕ are homotopic).

Let I be an assignment of an injective resolution to every object of A (we need a strong axiomof choice to show that such an assignment exists) and for fixed n ≥ 0 let RnT (A) denote theobject calculated with the injective resolution chosen by I. Let RnT (α) : RnT (A) −→ RnT (B)denote the morphism induced by α : A −→ B. Then it is easy to check that RnT : A −→ B is acovariant additive functor. If necessary we denote this functor by RnIT to indicate its dependenceon I.

Proposition 21. Let T : A −→ B be a covariant additive functor. If I and J are two assignmentsof injective resolutions to objects of A, then for n ≥ 0 there is a canonical natural equivalenceRnIT

∼= RnJ T .

Proof. For an object A the morphism 1A : A −→ A gives rise to cochain morphisms ξ : I −→ Jand η : J −→ I of the corresponding injective resolutions. Moreover, ξη and ηξ induce theidentity on A and so by Corollary 20 they must be homotopic to the identity cochain morphisms.Applying T and passing to cohomology we obtain an isomorphism RnIT (A) ∼= RnJ T (A) which iseasily checked to be natural in A.

12

Definition 8. Let T : A −→ B be a covariant additive functor. For n ≥ 0 the covariant additivefunctor RnT : A −→ B is called the n-th right derived functor of T . This functor dependson the injective resolutions chosen, but is independent of these choices up to canonical naturalequivalence.

Lemma 22. If T : A −→ B is left exact, then R0T and T are canonically naturally equivalent. IfT is exact then RnT ∼= 0 for n 6= 0.

Proof. Let I be an injective resolution of A. Then 0 −→ A −→ I0 −→ I1 is exact, so 0 −→TA −→ TI0 −→ TI1 is exact. Hence H0(TA) ∼= TA. This isomorphism is readily seen to benatural in A. It is clear that if T is exact all the higher right derived functors are zero.

Proposition 23. For an injective object I, RnT (I) = 0 for n ≥ 1 and R0T (I) ∼= T (I).

Proof. This follows immediately from the fact that the cochain complex I with Ii = 0 except forI0 = I is an injective resolution of I.

Proposition 24. Suppose we have an exact sequence in A

0 // A // I0 // · · · // In−1 // Inµ // C // 0 (6)

where I0, . . . , In are all injective and n ≥ 0. If B is a category of modules and T : A −→ B is leftexact then there is an exact sequence

T (In)T (µ) // T (C) // Rn+1T (A) // 0

Proof. Find an injective In+1 and a monomorphism e : C −→ In+1 and in the usual way continuethis process to produce an injective resolution I whose differential ∂n : In −→ In+1 is eµ. SinceT is left exact, the following diagram is commutative with exact rows

T (In)

T (µ)

��

+3 T (In)

T (∂n)

��

// 0 //

��

0

0 // T (C)T (e)

// T (In+1)T (∂n+1)

// T (In+2)

The Snake Lemma provides us with an exact sequence

0 −→ CokerT (µ) −→ CokerT (∂n) −→ T (In+2)

But of course Rn+1T (A) = KerT (∂n+1)/ImT (∂n) so the modules CokerT (µ) and Rn+1T (A)are isomorphic. The morphism T (C) −→ Rn+1T (A) fitting into the exact sequence is actuallyx 7→ T (e)(x) + ImT (∂n).

Suppose for some fixed n ≥ 0 we have an exact sequence of the form (6) for every object A ofA, which we can extend to an injective resolution. Define the additive functor rn+1T : A −→ Bas follows. The module rn+1T (A) is CokerT (µ). A morphism α : A −→ B can be lifted to acochain morphism which induces α′ : C −→ D in the following diagram

0 // A

α

��

// I0

��

// · · · // In−1 //

��

In

��

µ // C

α′

��

// 0

0 // B // J0 // · · · // Jn−1 // Jn τ// D // 0

Let rn+1T (α) : CokerT (µ) −→ CokerT (τ) be x+ ImT (µ) 7→ T (α′)(x) + ImT (τ). As we willsee in a moment, this is independent of the cochain morphism lifting α and defines an additivefunctor rn+1T .

13

If Rn+1T is defined using injective resolutions obtained from these exact sequences, then it isnot hard to see that there is a commutative diagram

T (In)T (µ) //

��

T (C)

%%KKKKKKKKKK

��

// CokerT (µ)

��

Rn+1T (A)

3;pppppppppp

pppppppppp

��

T (Jn)T (τ)

// T (D)

%%KKKKKKKKKK// CokerT (τ)

Rn+1T (B)

3;pppppppppp

pppppppppp

which shows that rn+1T is well-defined and is naturally equivalent to Rn+1T . So providedT : A −→ B is a left exact functor into a category of modules we can calculate the right de-rived functors R1T,R2T, . . . using finite exact sequences instead of injective resolutions.

Let A be an abelian category with enough projectives and T : A −→ B a contravariant additivefunctor. Then Aop is an abelian category with enough injectives, so for n ≥ 0 we can define theright derived functor RnT of T ∗, which is a contravariant additive functor RnT : A −→ B. Let Pbe an assignment of projective resolutions to objects of A. Then this is an assignment of injectiveresolutions to objects of Aop and we calculate RnT as follows: let P : · · · −→ P1 −→ P0 −→A −→ 0 be the projective resolution of A, and consider the cochain complex of objects of B

TI : 0 −→ T (P0) −→ T (P1) −→ T (P2) −→ · · ·

Then RnT (A) is the cohomology object Hn(TI). Given α : A −→ A′ let ϕ : P −→ P ′ be thecochain morphism inducing α. Then T (ϕ) : TP ′ −→ TP is a cochain morphism and RnT (α) :RnT (A′) −→ RnT (A) is HnT (ϕ). We call RnT the n-th right derived functor of T .

6 The Long Exact (Co)Homology Sequence

Throughout this section A is an abelian category.

Lemma 25. Let C be a chain complex in A. For n ∈ Z there is a unique morphism ∂n :Coker∂n+1 −→ Ker∂n−1 making the following diagram commute:

· · · // Cn+1∂n+1 // Cn

∂n //

��

Cn−1∂n−1 // Cn−2

// · · ·

Coker∂n+1// Ker∂n−1

OO

Moreover this morphism fits into an exact sequence

0 −→ Hn(C) −→ Coker∂n+1 −→ Ker∂n−1 −→ Hn−1(C) −→ 0 (7)

Proof. The morphism Cn −→ Im∂n is a quotient Cn/Ker∂n and Coker∂n+1 is a quotientCn/Im∂n+1 so the inclusion Im∂n+1 ≤ Ker∂n gives rise to an epimorphism Coker∂n+1 −→ Im∂n.Composed with Im∂n ≤ Ker∂n−1 this gives us the desired morphism.

14

Consider the following commutative diagram, in which the rows and first two columns areexact

0

��

0

��

0

��0 // Im∂n+1

��

// Ker∂n

��

// Hn(C) //

��

0

0 // Im∂n+1

��

// Cn

��

// Coker∂n+1//

��

0

0 // 0

��

// Im∂n

��

+3 Im∂n //

��

0

0 0 0

The Nine Lemma shows that the third column is also exact. Composing with the monomorphismIm∂n −→ Ker∂n−1 we see that (7) is exact at Hn(C) and Coker∂n+1. Since Ker∂n−1 −→Hn−1(C) is a cokernel of Im∂n −→ Ker∂n−1 composing with the epimorphism Coker∂n+1 −→Im∂n shows that the rest of the sequence is exact.

Theorem 26. Suppose we are given a short exact sequence of chain complexes in A

0 // Aϕ // B

ψ // C // 0

Then there is a canonical sequence of morphisms ωn : Hn(C) −→ Hn−1(A) for n ∈ Z called theconnecting morphisms with the property that the following is a long exact sequence

Hn+1(A)

��

Hn(A)

��

Hn−1(A)

��

<<

Hn+1(B)

��

Hn(B)

��

Hn−1(B)

��Hn+1(C)

BB�����������������Hn(C)

BB�����������������Hn−1(C)

<<

Proof. For n ∈ Z exactness of the sequences 0 −→ An −→ Bn −→ Cn −→ 0 gives rise to exactsequences 0 −→ Ker∂n −→ Ker∂n −→ Ker∂n and Coker∂n −→ Coker∂n −→ Coker∂n −→ 0(the first kernel being of ∂n : An −→ An−1, the second of ∂n : Bn −→ Bn−1 and so on), since takingkernels is left exact and taking cokernels is right exact. So for any n ∈ Z we get a commutative

15

diagram where the middle two rows and all columns are exact

0

��

0

��

0

��Hn(A)

��

Hn(ϕ) // Hn(B)

��

Hn(ψ) // Hn(C)

��Coker∂n+1

e∂n

��

// Coker∂n+1

e∂n

��

// Coker∂n+1//

e∂n

��

0

0 // Ker∂n−1//

��

Ker∂n−1

��

// Ker∂n−1

��Hn−1(A)

��

Hn−1(ϕ)// Hn−1(B)

Hn−1(ψ)//

��

Hn−1(C)

��0 0 0

(8)

Then by the Snake Lemma (which is valid in A by our Diagram Chasing notes) there is a canonicalmorphism ωn : Hn(C) −→ Hn−1(A) with the property that the following sequence is exact

Hn(A) // Hn(B) // Hn(C)ωn // Hn−1(A) // Hn−1(B) // Hn−1(C)

Piecing these exact sequences together gives the result.

The connecting morphism is canonical in the following sense: once we choose our canonicalkernels, cokernels and images for all the morphisms in the category, then (8) is canonically con-structed since it involves no choices (even the ∂ morphisms are unique with a certain property)and there is a canonical choice for ωn for each n (it has some unique property with respect tosome of the morphisms in (8)). So the connecting morphisms ωn depend only on the morphismsϕ,ψ and the canonical structures on A. If two people choose different canonical structures, thenthey may find different connecting morphisms (see Section 9).

Proposition 27. The connecting morphisms are natural with respect to the exact sequence. Givena commutative diagram in ChA with exact rows

0 // A

α

��

ϕ // B

β

��

ψ // C

γ

��

// 0

0 // A′ϕ′

// B′ψ′

// C ′ // 0

Then for every n ∈ Z there is a commutative diagram

Hn(A)

��

// Hn(B)

��

// Hn(C)

��

ω // Hn−1(A)

��

// Hn−1(B)

��

// Hn−1(C)

��Hn(A′) // Hn(B′) // Hn(C ′)

ω′// Hn−1(A′) // Hn−1(B′) // Hn−1(C ′)

where ω, ω′ are the respective connecting morphisms.

Proof. We need only check commutativity of the middle square. Using α, β, γ we induce morphismsbetween the cokernels and kernels to obtain a commutative diagram with exact rows of the type

16

given in our Diagram chasing result about naturality of the connecting morphism. This diagram isa “morphism” from the middle horizontal band of the diagrams (8) for A,B,C to the correspondingpiece of the diagram for A′, B′, C ′. Since the connecting morphism is natural with respect tosuch morphisms, it follows that the required diagram commutes (one checks that the morphismsHn(−) in our diagram above agree with the induced morphisms we get from the morphism ofdiagrams).

As a Corollary we see that the connecting morphism is in a sense independent of the middleterm of the sequence

Corollary 28. Suppose we have a commutative diagram of chain sequences in A with exact rows:

0 // A

1A

��

ϕ // B

γ

��

ψ // C //

1C

��

0

0 // A′ϕ′

// B′ψ′

// C ′ // 0

(9)

Then both sequences have the same connecting morphisms Hn(C) −→ Hn−1(A) for n ∈ Z.

The same proofs (just replacing subscripts by superscripts and massaging indices, which don’tplay any important role in the proof) work for cochains.

Theorem 29. Suppose we are given a short exact sequence of cochain complexes in A

0 // Aϕ // B

ψ // C // 0

Then there is a canonical sequence of morphisms ωn : Hn(C) −→ Hn+1(A) for n ∈ Z called theconnecting morphisms with the property that the following is a long exact sequence

Hn−1(A)

��

Hn(A)

��

Hn+1(A)

��

<<

Hn−1(B)

��

Hn(B)

��

Hn+1(B)

��Hn−1(C)

BB�����������������Hn(C)

BB�����������������Hn+1(C)

<<

Once again the connecting morphisms are canonical in the sense that they depend only on themorphisms ϕ,ψ and the canonical structures on A.

Proposition 30. The connecting morphisms are natural with respect to the exact sequence. Givena commutative diagram in coChA with exact rows

0 // A

α

��

ϕ // B

β

��

ψ // C

γ

��

// 0

0 // A′ϕ′

// B′ψ′

// C ′ // 0

Then for every n ∈ Z there is a commutative diagram

Hn(A)

��

// Hn(B)

��

// Hn(C)

��

ω // Hn+1(A)

��

// Hn+1(B)

��

// Hn+1(C)

��Hn(A′) // Hn(B′) // Hn(C ′)

ω′// Hn+1(A′) // Hn+1(B′) // Hn+1(C ′)

where ω, ω′ are the respective connecting morphisms.

17

Corollary 31. Suppose we have a commutative diagram of cochain sequences in A with exactrows:

0 // A

1A

��

ϕ // B

γ

��

ψ // C //

1C

��

0

0 // A′ϕ′

// B′ψ′

// C ′ // 0

(10)

Then both sequences have the same connecting morphisms Hn(C) −→ Hn+1(A) for n ∈ Z.

7 Two Long Exact Sequences

7.1 Left Derived Functors

Throughout this section we work in an abelian category A. For an object A a projective presen-tation of A is just an epimorphism P −→ A where P is projective. If we say ε : P −→ A is aprojective resolution of A we mean that the chain complex P together with ε : P0 −→ A is aprojective resolution.

Proposition 32. Given a short exact sequence

0 // A′ϕ // A

ψ // A′′ // 0

and projective presentations ε′ : P ′ −→ A′ and ε′′ : P ′′ −→ A′′ there is a projective presentationε : P −→ A fitting into a commutative diagram with exact rows

0 // P ′ι //

ε′

��

P

ε

��

π // P ′′ //

ε′′

��

0

0 // A′ ϕ// A

ψ// A′′ // 0

Proof. Let P = P ′ ⊕ P ′′ with ι, π the obvious morphisms. We define ε by components. The firstcomponent is ϕε′ and for the second we use the fact that P ′′ is projective to construct ξ : P ′′ −→ Asuch that ψξ = ε′′. Using the coproduct, it is clear that the diagram commutes. It follows from the5-Lemma for abelian categories (see our Diagram Chasing notes) that ε is an epimorphism.

Corollary 33. Let A be an abelian category with enough projectives. Given a short exact sequence

0 // A′ϕ // A

ψ // A′′ // 0

and projective resolutions ε′ : P ′ −→ A′ and ε′′ : P ′′ −→ A′′ there is a projective resolutionε : P −→ A and an exact sequence of chain complexes

0 // P ′ι // P

π // P ′′ // 0

with the property that ι, π induce ϕ,ψ respectively.

Proof. By the previous result we can construct a commutative diagram with exact rows andcolumns:

0 // P ′0 //

ε′

��

P0

ε

�� ��

// P ′′0

ε′′

��

//

��

0

0 // A′ ϕ//

��

A

��

ψ// A′′ //

��

0

0 0 0

18

By the Snake Lemma the sequence of kernels 0 −→ Kerε′ −→ Kerε −→ Kerε′′ −→ 0 is exact.Repeating this procedure with the sequence of kernels and proceeding recursively, we constructan exact sequence of complexes 0 −→ P ′ −→ P −→ P ′′ −→ 0 where P ′, P, P ′′ are projectiveresolutions of A′, A,A′′ respectively. Note that Pn = P ′n ⊕ P ′′n for n ≥ 0 and ιn, πn are theinjection and projection respectively, but the complex P is not necessarily the coproduct P ′⊕P ′′since the differentials may not be the coproduct differentials.

The resolution P is not unique, since its construction involved a lot of choices. But we canstill say a few things about any resolution produced by the Corollary. There is a morphismσ : P ′′0 −→ A such that ε = (ϕε′, σ) and morphisms λn : P ′′n −→ P ′n−1 for n ≥ 1 such that thedifferential is

∂n =(∂′n λn0 ∂′′n

)These morphisms satisfy the following relations:

ε′′ = ψσ

ϕε′λ1 + σ∂′′1 = 0∂′nλn+1 + λn∂

′′n+1 = 0 n ≥ 1

(11)

Theorem 34. Let A be an abelian category with enough projectives, B an abelian category, andT : A −→ B an additive functor. Suppose we have an exact sequence

0 // A′ϕ // A

ψ // A′′ // 0 (12)

Then there exist canonical connecting morphisms ωn : LnT (A′′) −→ Ln−1T (A′) for n ≥ 1 withthe property that the following sequence is exact

· · · // LnT (A′) // LnT (A) // LnT (A′′)ωn // Ln−1T (A′) // · · ·

· · · // L1T (A′′)ω1 // L0T (A′) // L0T (A) // L0T (A′′) // 0

(13)

Moreover these connecting morphisms are natural in the exact sequence: if we have a commutativediagram with exact rows

0 // A′ϕ //

f ′

��

A

f

��

ψ // A′′ //

f ′′

��

0

0 // B′α

// Bβ

// B′′ // 0

(14)

Then the following diagram commutes for all n ≥ 1

· · · // LnT (A′) //

��

LnT (A)

��

// LnT (A′′)

��

ωn // Ln−1T (A′)

��

// · · ·

· · · // LnT (B′) // LnT (B) // LnT (B′′)ωn

// Ln−1T (B′) // · · ·

Proof. Let P be an assignment of projective resolutions and assume the left derived functors are allcalculated with respect to P. Let ε′ : P ′ −→ A′, ε′′ : P ′′ −→ A′′ be arbitrary projective resolutions(not necessarily from P) and let ε : P −→ A be a resolution constructed by the technique of theCorollary. Since Pn = P ′n ⊕ P ′′n for n ∈ Z and T is additive, it follows that the sequence

0 −→ TP ′ −→ TP −→ TP ′′ −→ 0 (15)

is also short exact. Then Theorem 26 yields morphisms ω′n : Hn(TP ′′) −→ Hn−1(TP ′) which fitinto an exact sequence of the homology objects of (15). As in the proof of Proposition 14 there

19

are canonical isomorphisms LnT (A′′) ∼= Hn(TP ′′) and Ln−1T (A′) ∼= Hn−1(TP ′) for n ≥ 1, andwe define ωn to be the unique morphism making the following diagram commute

LnT (A′′)

��

ωn // Ln−1T (A′)

��Hn(TP ′′)

ω′n

// Hn−1(TP ′)

Then it is clear that the ωn make the sequence (13) exact (it is not clear at this point thatωn is independent of the choice of the resolutions P ′, P, P ′′). Next we prove naturality of theseconnecting morphisms. Suppose we are given projective resolutions ε′ : P ′ −→ A′, ε′′ : P ′′ −→A′′, η′ : Q′ −→ B′ and η′′ : Q′′ −→ B′′ and a commutative diagram with exact rows (14).Let ε : P −→ A and η : Q −→ B be projective resolutions produced by the Corollary. LetF ′ : P ′ −→ Q′ lift f ′ and F ′′ : P ′′ −→ Q′′ lift f ′′. We claim there is a chain morphism F : P −→ Qlifting f and giving a commutative diagram with exact rows

0 // P ′ //

F ′

��

P //

F

��

P ′′ //

F ′′

��

0

0 // Q′ // Q // Q′′ // 0

(16)

In order to produce F , we will construct for n ≥ 0 morphisms γn : P ′′n −→ Q′n (not a chainmorphism) and then set

Fn =(F ′n γn0 F ′′n

): P ′n ⊕ P ′′n −→ Q′n ⊕Q′′n

For n ≥ 0 we let u′n : P ′n −→ Pn, p′n : Pn −→ P ′n and u′′n : P ′′n −→ Pn, p

′′n : Pn −→ P ′′n be

the respective injections and projections, and similarly we use the notation v′n, q′n, v

′′n, q

′′n for the

product Qn. We know there are morphisms σP : P ′′0 −→ A, σQ : Q′′0 −→ B and λn : P ′′n −→P ′n−1, µn : Q′′n −→ Q′n−1 which satisfy the following relations for n ≥ 1

∂Pn =(∂′n λn0 ∂′′n

)∂Qn =

(∂′n µn0 ∂′′n

)ε′′ = ψσP η′′ = βσQ

ϕε′λ1 + σP∂′′1 = 0 αη′µ1 + σQ∂′′1 = 0∂′nλn+1 + λn∂

′′n+1 = 0 ∂′nµn+1 + µn∂

′′n+1 = 0

Also ε = (ϕε′, σP ), η = (αη′, σQ) so σP = εu′′0 , σQ = ηv′′0 . Suppose that F is a lifting of f , so the

map ηF0 − fε from P0 = P ′0 ⊕ P ′′0 is zero. Writing out the matrices we see that we must haveαη′γ0 = fσP − σQF ′′0 .

Assuming nothing about F we use the above to motivate the following definition of γ0: onechecks easily that β(fσP − σQF ′′0 ) = 0 so there is a unique morphism τ : P ′′0 −→ B′ withατ = fσP − σQF ′′0 . Use projectivity of P ′′0 to lift τ to a morphism γ0 : P ′′0 −→ Q′0 with theproperty that αη′γ0 = fσP − σQF ′′0 . So we have constructed γ0 with ηF0 = fε.

For F to be a chain morphism we must have dQF = FdP . Expanding this out shows that wehave to construct morphisms γn : P ′′n −→ Q′n for n ≥ 1 with the property that ∂′nγn = gn where

gn = γn−1∂′′n − µnF

′′n + F ′n−1λn

The morphism g1 : P ′′1 −→ Q′0 satisfies η′g1 = 0, since

αη′g1 = αη′γ0∂′′1 + αη′F ′0λ1 − αη′µ1F

′′1

= (fσP − σQF ′′0 )∂′′1 + αη′F ′0λ1 − αη′µ1F′′1

= fσP∂′′1 − σQF ′′0 ∂′′1 + αη′F ′0λ1 − αη′µ1F

′′1

20

But σQF ′′0 ∂′′1 = σQ∂′′1F

′′1 = −αη′µ1F

′′1 and fσP∂′′1 = −fϕε′λ1 = −αη′F ′0λ1 so it is clear that

αη′g1 = 0 and therefore η′g1 = 0. It follows that g1 factors through Im(Q′′1 −→ Q′′0) and we canuse projectivity of P ′′1 to lift to a morphism γ1 : P ′′1 −→ Q′′0 with ∂′1γ1 = g1, as required. Giventhat we have constructed γn with ∂′nγn = gn it is not hard to show that ∂′n+1gn+1 = 0, so we canlet γn be any lift of the factorisation of gn+1 through Ker∂′n+1 = Im∂′n+2. We have constructedγn for all n, and it is clear that the morphism F thus defined has the required properties.

Using this fact and Proposition 27 in the case where both rows of (14) are the same andthe vertical morphisms are identities, we see that the morphism ωn : LnT (A′′) −→ Ln−1T (A′)doesn’t depend on the choices leading to the construction of the exact sequence 0 −→ TP ′ −→TP −→ TP ′′ −→ 0. That is, the connecting morphism depends only on the morphisms ψ,ϕ, theassignment of resolutions P and the canonical structures on B. The naturality of the connectingmorphisms ωn for the left derived functors follows from the naturality of the connecting morphismfor homology and the diagram (16).

Corollary 35. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B an additive functor. Then the functor L0T : A −→ B is right exact.

The connecting morphisms are also independent of the choice of resolutions P, in the followingsense: if Q is another assignment of resolutions then, with the vertical isomorphisms canonical,there is a commutative diagram for all n ≥ 1

· · · // LPn T (A′) //

��

LPn T (A) //

��

LPn T (A′′)

��

ωPn // LPn−1T (A′)

��

// · · ·

· · · // LQn T (A′) // LQn T (A) // LQn T (A′′)ωQn

// LQn−1T (A′) // · · ·

We have proved naturality of the sequence (13) in the objects A. It is also natural in the functor.

Definition 9. Let A be an abelian category with enough projectives, B an abelian category andτ : T −→ T ′ a natural transformation of additive covariant functors T, T ′ : A −→ B. Then forany chain complex X there is a chain morphism τX : TX → T ′X defined by (τX)n = τXn

. Inparticular if P is an assignment of projective resolutions and if P is the resolution of A thenwe obtain a chain map τP : TP −→ T ′P . Taking homology gives a natural transformation ofthe left derived functors Lnτ : LnT −→ LnT

′ with (Lnτ)A = Hn(τP ) for n ≥ 0. Notice thatLn(τρ) = (Lnτ)(Lnρ), Ln(τ + ρ) = Ln(τ) + Ln(ρ) and Ln(1T ) = 1LnT .

Lemma 36. Let A be an abelian category with enough projectives, B an abelian category, andT, T ′ : A −→ B right exact functors. Then the following diagram commutes for any naturaltransformation τ : T −→ T ′

T

��

τ // T ′

��L0T

L0τ// L0T

′

Proposition 37. Let A be an abelian category with enough projectives, B an abelian category,and τ : T −→ T ′ a natural transformation of additive functors T, T ′ : A −→ B. Suppose we havean exact sequence

0 // A′ϕ // A

ψ // A′′ // 0 (17)

Then the following diagram is commutative, where the connecting morphisms are canonical

· · · // LnT (A′) //

(Lnτ)A′

��

LnT (A) //

(Lnτ)A

��

LnT (A′′)

(Lnτ)A′′

��

ωn// Ln−1T (A′)

(Lnτ)A′

��

// · · ·

· · · // LnT ′(A′) // LnT ′(A) // LnT ′(A′′)ωn

// Ln−1T′(A′) // · · ·

21

Proof. Let P be an assignment of projective resolutions with respect to which all derived functorsare calculated. Let P ′, P ′′ be the assigned resolutions of A′, A′′ respectively and construct in theusual way a resolution P fitting into an exact sequence 0 −→ P ′ −→ P −→ P ′′ −→ 0. There is acommutative diagram with exact rows

0 // TP ′

τP ′

��

// TP

τP

��

// TP ′′

τP ′′

��

// 0

0 // T ′P ′ // T ′P // T ′P ′′ // 0

Since (Lnτ)A′′ = Hn(τP ′′) and (Lnτ)A′ = Hn(τP ′) the commutativity of the required diagramfollows immediately from the construction of the connecting morphisms in Theorem 34 and nat-urality of the connecting morphism of chain sequences in B with respect to diagrams like the oneabove.

Definition 10. Let A,B be abelian categories. A sequence T ′ −→ T −→ T ′′ of additive functorsA −→ B and natural transformations is called exact on projectives if for every projective objectP the sequence 0 −→ T ′(P ) −→ T (P ) −→ T ′′(P ) −→ 0 is exact.

Proposition 38. Let A be an abelian category with enough projectives, B an abelian category,and suppose there is a sequence of additive functors A −→ B which is exact on projectives

T ′τ // T

ρ // T ′′

Then for every object A there are canonical connecting morphisms ωn : LnT ′′(A) −→ Ln−1T′(A)

for n ≥ 1 with the property that the following sequence is exact

· · · // LnT ′(A)(Lnτ)A// LnT (A)

(Lnρ)A// LnT ′′(A)ωn // Ln−1T

′(A) // · · ·

· · · // L1T′′(A)

ω1 // L0T′(A)

(L0τ)A // L0T (A)(L0ρ)A// L0T

′′(A) // 0

(18)

This sequence is natural in both A and the exact sequence. For any morphism α : A −→ B thefollowing diagram is commutative

· · · // LnT ′(A) //

��

LnT (A) //

��

LnT′′(A)

��

ωn // Ln−1T′(A) //

��

· · ·

· · · // LnT ′(B) // LnT (B) // LnT ′′(B)ωn

// Ln−1T′(B) // · · ·

(19)

and for any commutative diagram of additive functors with rows exact on projectives

T ′

ϕ′

��

τ // T

ϕ

��

ρ // T ′′

ϕ′′

��S′ σ

// Sθ

// S′′

(20)

the following diagram is commutative for any object A

· · · // LnT ′(A) //

(Lnϕ′)A

��

LnT (A) //

(Lnϕ)A

��

LnT′′(A)

ωn //

(Lnϕ′′)A

��

Ln−1T′(A) //

(Ln−1ϕ′)A

��

· · ·

· · · // LnS′(A) // LnS(A) // LnS′′(A)ωn

// Ln−1S′(A) // · · ·

(21)

22

Proof. Let P be an assignment of projective resolutions with respect to which all derived functorsare calculated. Let P be the assigned resolution of A. Since the sequence of functors is exact onprojectives the following sequence is exact

0 −→ T ′P −→ TP −→ T ′′P −→ 0

The long exact homology sequence then yields the connecting morphisms ωn and exactness of(18). Given a morphism α : A −→ B let Q be the resolution of B and lift α to a morphism ofchain complexes ϕ : P −→ Q. There is a commutative diagram with exact rows

0 // T ′P

T ′ϕ

��

// TP //

Tϕ

��

T ′′P //

T ′′ϕ

��

0

0 // T ′Q // TQ // T ′′Q // 0

Naturality of the connecting morphism for homology with respect to such diagrams shows that(19) commutes. Commutativity of (21) follows similarly.

The connecting morphisms depend only on τ, ρ, the canonical structures on B and the assign-ment of projective resolutions P used to calculate the left derived functors. In fact, they are alsoindependent of the choice of resolutions P, in the following sense: if Q is another assignment ofresolutions then, with the vertical isomorphisms canonical, there is a commutative diagram for alln ≥ 1

· · · // LPn T′(A) //

��

LPn T (A) //

��

LPn T′′(A)

��

ωPn // Ln−1T′(A)

��

// · · ·

· · · // LQn T′(A) // LQn T (A) // LQn T

′′(A)ωQn

// LQn−1T′(A) // · · ·

7.2 Right Derived Functors

For an object A an injective presentation of A is just a monomorphism A −→ I where I is injective.If we say ε : A −→ I is an injective resolution of A we mean that the cochain complex I togetherwith ε : A −→ I0 is an injective resolution.

Proposition 39. Given a short exact sequence

0 // A′ϕ // A

ψ // A′′ // 0

and injective presentations ε′ : A′ −→ I ′′ and ε′′ : A′′ −→ I ′′ there is an injective presentationε : A −→ I fitting into a commutative diagram with exact rows

0 // I ′ι // I

π // I ′′ // 0

0 // A′

ε′

OO

ϕ// A

ε

OO

ψ// A′′

ε′′

OO

// 0

Proof. Let I = I ′ ⊕ I ′′ with ι, π the obvious morphisms. We define ε by components. The secondcomponent is ε′′ψ and for the first we use the fact that I ′ is injective to construct ξ : A −→ I ′

such that ξϕ = ε′. Using the product, it is clear that the diagram commutes. It follows from the5-Lemma for abelian categories (see our Diagram Chasing notes) that ε is a monomorphism.

Corollary 40. Let A be an abelian category with enough injectives. Given a short exact sequence

0 // A′ϕ // A

ψ // A′′ // 0

23

and injective resolutions ε′ : A′ −→ I ′ and ε′′ : I ′′ −→ A′′ there is an injective resolutionε : A −→ I and an exact sequence of cochain complexes

0 // I ′ι // I

π // I ′′ // 0

with the property that ι, π induce ϕ,ψ respectively.

Proof. The proof follows from the previous result just as in the chain case. The (nonunique)resolution produced can be described as follows (we use subscripts to avoid bad notation): for alln ≥ 0, In = I ′n ⊕ I ′′n and ιn, πn are part of this biproduct. There is a morphism σ : A −→ I ′0 andmorphisms λn : I ′′n −→ I ′n+1 for n ≥ 0 such that

ε =(σε′′ψ

)∂n =

(∂′n λn0 ∂′′n

)These morphisms satisfy the following relations for n ≥ 0

ε′ = σϕ

∂′0σ + λ0ε′′ψ = 0

∂′n+1λn + λn+1∂′′n = 0

Theorem 41. Let A be an abelian category with enough injectives, B an abelian category, and letT : A −→ B be an additive functor. Suppose we have an exact sequence

0 // A′ϕ // A

ψ // A′′ // 0 (22)

Then there exist canonical connecting morphisms ωn : RnT (A′′) −→ Rn+1T (A′) for n ≥ 0 withthe property that the following sequence is exact

0 // R0T (A′) // R0T (A) // R0T (A′′) ω0// R1T (A′) // · · ·

· · · // RnT (A′′) ωn// Rn+1T (A′) // Rn+1T (A) // Rn+1T (A′′) // · · ·

(23)

Moreover these connecting morphisms are natural in the exact sequence: if we have a commutativediagram with exact rows

0 // A′ϕ //

f ′

��

A

f

��

ψ // A′′ //

f ′′

��

0

0 // B′α

// Bβ

// B′′ // 0

(24)

Then the following diagram commutes for all n ≥ 0

· · · // RnT (A′) //

��

RnT (A)

��

// RnT (A′′)

��

ωn// Rn+1T (A′)

��

// · · ·

· · · // RnT (B′) // RnT (B) // RnT (B′′)ωn

// Rn+1T (B′) // · · ·

Proof. We just follow the proof of the chain case, making suitable modifications. We assumethe right derived functors are all calculated using some assignment I of injective resolutions. Tocalculate the connecting morphisms you take any injective resolutions ε′ : A′ −→ I ′, ε′′ : A′′ −→I ′′, construct ε : A −→ I and use the connecting morphism of the long exact cohomology sequence

24

arising from 0 −→ TI ′ −→ TI −→ TI ′′ −→ 0, together with the isomorphisms of RnT (A′)and RnT (A′′) with the suitable cohomology objects in this sequence. This is independent of theresolutions I ′, I, I ′′ you use, and naturality follows easily. The connecting morphisms are canonicalin the sense that they depend only on the morphisms ϕ,ψ, the assignment of resolutions I andthe canonical structures on B.

Corollary 42. Let A be an abelian category with enough injectives, B an abelian category andT : A −→ B an additive functor. Then the functor R0T : A −→ B is left exact.

The connecting morphisms are also independent of the choice of resolutions I, in the followingsense: if J is another assignment of resolutions then, with the vertical isomorphisms canonical,there is a commutative diagram for all n ≥ 0

· · · // RnIT (A′) //

��

RnIT (A) //

��

RnIT (A′′)ωnI //

��

Rn+1I T (A′)

��

// · · ·

· · · // RnIT (A′) // RnIT (A) // RnJ T (A′′)ωnJ

// Rn+1J T (A′) // · · ·

We have proved naturality of the sequence (23) in the objects A. It is also natural in thefunctor.

Definition 11. Let A be an abelian category with enough injectives, B an abelian category andτ : T −→ T ′ a natural transformation of additive covariant functors T, T ′ : A −→ B. Then forany cochain complex Y there is a cochain morphism τY : TY −→ T ′Y defined by (τY )n = τYn .In particular if I is an assignment of injective resolutions and if I is the resolution of A thenwe obtain a cochain map τI : TI −→ T ′I. Taking cohomology gives a natural transformation ofthe right derived functors Rnτ : RnT −→ RnT ′ with (Rnτ)A = Hn(τI) for n ≥ 0. Notice thatRn(τρ) = (Rnτ)(Rnρ), Rn(τ + ρ) = Rn(τ) +Rn(ρ) and Rn(1T ) = 1RnT .

Lemma 43. Let A be an abelian category with enough injectives, B an abelian category, and T, T ′ :A −→ B left exact functors. Then the following diagram commutes for any natural transformationτ : T −→ T ′

T

��

τ // T ′

��R0T

R0τ

// R0T ′

Proposition 44. Let A be an abelian category with enough injectives, B an abelian category andτ : T −→ T ′ a natural transformation of additive functors T, T ′ : A −→ B. Suppose we have anexact sequence

0 // A′ϕ // A

ψ // A′′ // 0 (25)

Then the following diagram is commutative, where the connecting morphisms are canonical

· · · // RnT (A′) //

(Rnτ)A′

��

RnT (A) //

(Rnτ)A

��

RnT (A′′)

(Rnτ)A′′

��

ωn// Rn+1T (A′)

(Rnτ)A′

��

// · · ·

· · · // RnT ′(A′) // RnT ′(A) // RnT ′(A′′)ωn

// Rn+1T ′(A′) // · · ·

Definition 12. Let A,B be abelian categories. A sequence T ′ −→ T −→ T ′′ of additive functorsA −→ B and natural transformations is called exact on injectives if for every injective object Ithe sequence 0 −→ T ′(I) −→ T (I) −→ T ′′(I) −→ 0 is exact.

25

Proposition 45. Let A be an abelian category with enough injectives, B an abelian category andsuppose there is a sequence of additive functors A −→ B which is exact on injectives

T ′τ // T

ρ // T ′′

Then for every object A there are canonical connecting morphisms ωn : RnT ′′(A) −→ Rn+1T ′(A)for n ≥ 0 with the property that the following sequence is exact

0 // R0T ′(A)(R0τ)A // R0T (A)

(R0ρ)A// R0T ′′(A) ω0// R1T ′(A) // · · ·

· · · // RnT ′(A)(Rnτ)A// RnT (A)

(Rnρ)A// RnT ′′(A) ωn// Rn+1T ′(A) // · · ·

This sequence is natural in both A and the exact sequence. For any morphism α : A −→ B thefollowing diagram is commutative

· · · // RnT ′(A) //

��

RnT (A) //

��

RnT ′′(A)

��

ωn// Rn+1T ′(A) //

��

· · ·

· · · // RnT ′(B) // RnT (B) // RnT ′′(B)ωn

// Rn+1T ′(B) // · · ·

(26)

and for any commutative diagram of additive functors with rows exact on injectives

T ′

ϕ′

��

τ // T

ϕ

��

ρ // T ′′

ϕ′′

��S′ σ

// Sθ

// S′′

(27)

the following diagram is commutative for any object A

· · · // RnT ′(A) //

(Rnϕ′)A

��

RnT (A) //

(Rnϕ)A

��

RnT ′′(A) ωn//

(Rnϕ′′)A

��

Rn+1T ′(A) //

(Rn+1ϕ′)A

��

· · ·

· · · // RnS′(A) // RnS(A) // RnS′′(A)ωn

// Rn+1S′(A) // · · ·

(28)

Proof. Let I be an assignment of projective resolutions with respect to which all derived functorsare calculated. Let I be the assigned resolution of A. Since the sequence of functors is exact oninjectives the following sequence is exact

0 −→ T ′I −→ TI −→ T ′′I −→ 0

The long exact cohomology sequence then yields the connecting morphisms ωn and the variousother claims follow in the same way as for left derived functors.

The connecting morphisms depend only on τ, ρ, the canonical structures on B and the as-signment of injective resolutions I used to calculate the right derived functors. They are alsoindependent of the choice of resolutions I, in the following sense: if J is another assignment ofresolutions then, with the vertical isomorphisms canonical, there is a commutative diagram for alln ≥ 0

· · · // RnIT′(A)

��

// RnIT (A)

��

// RnIT′′(A)

��

ωnI // Rn+1T ′(A)

��

// · · ·

· · · // RnJ T′(A) // RnJ T (A) // RnJ T

′′(A)ωnJ

// Rn+1J T ′(A) // · · ·

26

8 Dimension Shifting

Definition 13. Let A be an abelian category with enough projectives, B an abelian category,and T : A −→ B an additive functor. An object Q is called left T -acyclic if LiT (Q) = 0 for alli ≥ 1. If there is no chance of confusion we say that Q is left acyclic or even just acyclic. It isclear that projective objects are acyclic.

Proposition 46. Let A be an abelian category with enough projectives, B an abelian category,and T : A −→ B an additive functor. Suppose we have an exact sequence in A with P projective

0 −→M −→ P −→ A −→ 0

Then the canonical connecting morphism ωn : LnT (A) −→ Ln−1T (M) is an isomorphism for alln ≥ 2. If T is right exact there is an exact sequence

0 −→ L1T (A) −→ T (M) −→ T (P )

Proof. More generally assume that P is left T -acyclic. The first claim follows immediately fromexactness of the long exact sequence of derived functors. If T is right exact, let L1T (A) −→T (M) be the canonical connecting morphism L1T (A) −→ L0T (M) followed by the canonicalisomorphism L0T (M) ∼= T (M). Then it is clear that the sequence given above is exact.

More generally, we have

Proposition 47 (Dimension Shifting). Let A be an abelian category with enough projectives,B an abelian category, and T : A −→ B an additive functor. Suppose we have an exact sequencein A with all Pi projective and m ≥ 0

0 −→M −→ Pm −→ Pm−1 −→ · · · −→ P0 −→ A −→ 0 (29)

Then there are canonical isomorphisms ρn : LnT (A) −→ Ln−m−1T (M) for n ≥ m + 2, and if Tis right exact there is an exact sequence

0 −→ Lm+1T (A) −→ T (M) −→ T (Pm) (30)

Both the isomorphisms ρn and the exact sequence (30) are natural in T , in the sense that for anatural transformation τ : T −→ T ′, n ≥ m+ 2 and m ≥ 0 the following two diagrams commute

LnT (A)

��

ρn +3 Ln−m−1T (M)

��LnT

′(A)ρn

+3 Ln−m−1T′(M)

0 // Lm+1T (A)

��

// T (M)

��

// T (Pm)

��0 // Lm+1T

′(A) // T ′(M) // T ′(Pm)

Proof. More generally we can assume that the Pi are left T -acyclic. If m = 0 then we are in thesituation of the previous Proposition, and we let ρn be the connecting morphism. If m ≥ 1 thenfor 0 ≤ i ≤ m− 1 let Ki −→ Pi be an image of Pi+1 −→ Pi. We have exact sequences

0 −→ K0 −→ P0 −→ A −→ 00 −→ K1 −→ P1 −→ K0 −→ 0

...0 −→ Km−1 −→ Pm−1 −→ Km−2 −→ 0

0 −→M −→ Pm −→ Km−1 −→ 0

Putting together the connecting morphisms for all these sequences gives an isomorphism ρn :LnT (A) −→ Ln−m−1T (M) for n ≥ m + 2, which depends only on the canonical structureson A and B. If T is right exact and m ≥ 1 let Lm+1T (A) −→ T (M) be the composite

27

Lm+1T (A) ∼= LmT (K0) ∼= · · · ∼= L1T (Km−1) −→ L0T (M) ∼= T (M). This clearly fits intothe required exact sequence. Naturality of the morphisms ρn and the sequence (30) follows imme-diately from Proposition 37. It is not difficult to check that the isomorphisms ρn and morphismLm+1T (A) −→ T (M) do not depend on the images Ki chosen, so they depend only on the as-signment of resolutions used to calculate the derived functors, the canonical structures on B, andthe exact sequence (29).

The morphisms ρn and Lm+1T (A) −→ T (M) are also independent of the assignment of resolu-tions used to calculate the left derived functors, in the following sense: if P,Q are two assignmentsof projective resolutions, and we are in the situation of the Proposition, then it is not hard tocheck that the following diagrams commute for n ≥ m+ 2 (vertical isomorphisms canonical)

LPn T (A)

��

ρn +3 LPn−m−1T (M)

��LQn T (A) ρn

+3 LQn−m−1T (M)

LPm+1T (A)

��

// T (M)

LQm+1T (A) // T (M)

The next result shows that Proposition 47 is natural in the exact sequence.

Proposition 48. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B an additive functor. Suppose we have a commutative diagram in A with exact rows,all Pi, Qi left T -acyclic and m ≥ 0

0 // M

β

��

// Pm

ψm

��

// Pm−1

ψm−1

��

// · · · // P0

ψ0

��

// A

α

��

// 0

0 // N // Qm // Qm−1// · · · // Q0

// B // 0

Then we claim that the morphisms of Proposition 47 fit into the following commutative diagramsfor n ≥ m+ 2

LnT (A)

LnT (α)

��

+3 Ln−m−1T (M)

Ln−m−1T (β)

��LnT (B) +3 Ln−m−1T (N)

0 // Lm+1T (A)

Lm+1T (α)

��

// T (M)

T (β)

��

// T (Pm)

T (ψm)

��0 // Lm+1T (B) // T (N) // T (Qm)

Proof. Both statements follow easily from the naturality of the connecting morphism with respectto morphisms of exact sequences (note that the right hand diagram only makes sense for T rightexact).

Let A be an abelian category with enough projectives, and let B be a category of modules. IfT : A −→ B is right exact then Propositions 47 and 17 both produce exact sequences

0 −→ Ln+1T (A) −→ T (M) −→ T (P )

from an exact sequence

0 −→M −→ Pm −→ · · · −→ P0 −→ A −→ 0

with all Pi projective. In Proposition 47 the morphism Ln+1T (A) −→ T (M) was defined as thecomposite of connecting morphisms, whereas in Proposition 17 we gave the map explicitly. Itwould be nice to know that these two maps are the same, which is what we now prove. We beginwith a technical result that does most of the work.

28

Proposition 49. Let A be an abelian category with enough projectives, B a category of modules,and T : A −→ B a right exact functor. Let A be an object of A with projective resolution· · · −→ P1 −→ P0 −→ A −→ 0 and let e : P1 −→ K,µ : K −→ P0 be an epi-mono factorisation of∂1 = µe with A 6= K, so we have an exact sequence

0 // Kµ // P0

// A // 0

If we choose the projective resolution · · · −→ P2 −→ P1 −→ K −→ 0 then the connecting morphismω1 : L1T (A) −→ L0T (K) is the canonical injection KerT (∂1)/ImT (∂2) −→ T (P1)/ImT (∂2)given by x+ ImT (∂2) 7→ x+ ImT (∂2) and for n > 1 the connecting morphism ωn is the identity.

Proof. All derived functors are calculated relative to the assignment P of projective resolutionswhich chooses the given resolution for A and the resolution · · · −→ P2 −→ P1 −→ K −→ 0 for K,with P1 −→ K being e. As in Corollary 33 we produce a resolution T of P0 with Tn = Pn+1 ⊕Pnfor n ≥ 0. The differentials ∂n : Tn −→ Tn−1 are constructed as follows. The epimorphismT0 −→ P0 is (∂1, 1). Let K be a pullback of ∂1 : P1 −→ P0 and 1 : P0 −→ P0. Then the inducedmorphism K −→ T0 is a kernel for T0 −→ P0. Similar arguments for higher n show that we canarrange for the differentials for n ≥ 1 to be of the form

∂n =(∂Kn λn0 ∂An

)(31)

where λn : Pn −→ Pn is the identity. The morphism ω1 is the connecting morphism of thefollowing commutative diagram with exact rows and columns

CokerT (∂K2 ) //

��

CokerT (∂2)

��

// CokerT (∂A2 )

��

// 0

0 // T (P1) // T (P1 ⊕ P0) // T (P0)

We can identify the module T (P1 ⊕ P0) with T (P1)⊕ T (P0) and similarly for T (P2 ⊕ P1). Givenx+ImT (∂A2 ) ∈ L1T (A) the image in CokerT (∂A2 ) is just x+ImT (∂2). We can choose the preimagein CokerT (∂2) to be (0, x) + ImT (∂2), which maps to (x, T (∂A1 )(x)) = (x, 0) ∈ T (P1 ⊕ P0) sinceby assumption x ∈ KerT (∂A1 ). Then we can choose the preimage x ∈ T (P1), which shows thatω1 is the desired map. A similar argument shows that the higher connecting morphisms areidentities.

Corollary 50. Let A be an infinite abelian category with enough projectives, B a category ofmodules, and T : A −→ B a right exact functor. Suppose we have an exact sequence in A with allPi projective and m ≥ 0

0 −→M −→ Pm −→ Pm−1 −→ · · · −→ P0 −→ A −→ 0

Assume that the resolution P chosen for A ends with Pm −→ · · · −→ P0 −→ A −→ 0 and lete : Pm+1 −→M be the unique factorisation of Pm+1 −→ Pm through M −→ Pm. Then

(i) The morphism Lm+1T (A) −→ T (M) is given by x+ ImT (∂m+2) 7→ T (e)(x).

(ii) If the resolution chosen for M is the one ending in e : Pm+1 −→ M obtained from P , thenfor n ≥ m+ 2 the isomorphism ρn : LnT (A) −→ Ln−m−1T (M) is the identity map.

Proof. (i) By induction on m. If m = 0 then L1T (A) −→ T (M) is the connecting morphismL1T (A) −→ L0T (M) followed by the canonical isomorphism L0T (M) ∼= T (M). Using the previ-ous Lemma it is easy to check this has the required form (we can assume M 6= A by replacing Mby an isomorphic copy and using naturality of the connecting morphism in the exact sequence).For m ≥ 1 choose an image K0 −→ P0 of P1 −→ P0 with K0 6= A. So we have exact sequences

0 −→ K0 −→ P0 −→ A −→ 00 −→M −→ Pm −→ · · · −→ P1 −→ K0 −→ 0

29

Let P be an assignment of resolutions which chooses a resolution for A ending in Pm −→ · · · −→P0 −→ A −→ 0 and chooses the resolution for K0 ending in Pm −→ · · · −→ P1 −→ K0 −→ 0obtained from the resolution of A. Let e be as in the statement of the Corollary. The canonicalmorphism Lm+1T (A) −→ T (M) is the composite of the connecting isomorphism Lm+1T (A) ∼=LmT (K0) with the morphism LmT (K0) −→ T (M) defined for the second exact sequence above.So by the inductive hypothesis and Proposition 49 the morphism Lm+1T (A) −→ T (M) has thedesired form. Since this morphism is independent of the choice of K0, the result holds for anyassignment of resolutions P which chooses a resolution for A ending in Pm −→ · · · −→ P0 −→A −→ 0 (it need not choose a special resolution for any particular K0). The proof of (ii) issimilar.

Definition 14. Let A be an abelian category with enough injectives, B an abelian category, andT : A −→ B an additive functor. An object Q is called right T -acyclic if RiT (Q) = 0 for all i ≥ 1.If there is no chance of confusion we say that Q is right acyclic or even just acyclic. It is clearthat injective objects are right acyclic.

The dual version of Proposition 47 is the following

Proposition 51. Let A be an abelian category with enough injectives, B an abelian category, andT : A −→ B an additive functor. Suppose we have an exact sequence in A with all Ii injectiveand m ≥ 0

0 −→ A −→ I0 −→ · · · −→ Im−1 −→ Im −→M −→ 0

Then there are canonical isomorphisms ρn : RnT (M) −→ Rn+m+1T (A) for n ≥ 1, and if T is leftexact there is an exact sequence

T (Im) −→ T (M) −→ Rm+1T (A) −→ 0 (32)

Both the isomorphisms ρn and the exact sequence (32) are natural in T , in the sense that for anatural transformation τ : T −→ T ′, n ≥ 1 and m ≥ 0 the following two diagrams commute

RnT (M)

��

ρn

// Rn+m+1T (A)

��RnT ′(M)

ρn// Rn+m+1T ′(A)

T (Im) //

��

T (M) //

��

Rm+1T (A) //

��

0

T ′(Im) // T ′(M) // Rm+1T ′(A) // 0

Proof. We work more generally with right T -acyclic Ii and proceed in the same way as for leftderived functors. Take canonical images and form a list of exact sequences

0 −→ Km −→ Im −→M −→ 0

0 −→ Km−1 −→ Im−1 −→ Km −→ 0...

0 −→ K1 −→ I1 −→ K2 −→ 0

0 −→ A −→ I0 −→ K1 −→ 0

For n ≥ 1 the connecting morphisms for all these sequences are isomorphisms, so we can take thecomposite RnT (M) ∼= Rn+1T (Km) ∼= · · · ∼= Rn+m+1T (A). For m = 0, ρn is just the connectingmorphism of 0 −→ A −→ I0 −→ M −→ 0, and the exact sequence (32) involves the morphismT (M) −→ R1T (A) given by the canonical isomorphism T (M) ∼= R0T (M) followed by the con-necting morphism. For m ≥ 1 we take T (M) ∼= R0T (M) followed by R0T (M) −→ R1T (Km) andthen a sequence of connecting isomorphisms R1T (Km) ∼= · · · ∼= Rm+1T (A). Naturality is easy tocheck.

30

Proposition 52. Let A be an abelian category with enough injectives, B an abelian category andT : A −→ B an additive functor. Suppose we have a commutative diagram in A with exact rows,all Ii, J i right T -acyclic and m ≥ 0

0 // A

α

��

// I0

ψ0

��

// · · · // Im−1 //

ψm−1

��

Im

ψm

��

// M

β

��

// 0

0 // B // J0 // · · · // Jm−1 // Jm // N // 0

Then we claim that the morphisms of Proposition 51 fit into the following commutative diagramsfor n ≥ 1

RnT (M)

RnT (β)

��

+3 Rn+m+1T (A)

Rn+m+1T (α)

��RnT (N) +3 Rn+m+1T (B)

T (Im)

T (ψm)

��

// T (M)

T (β)

��

// Rm+1T (A)

Rm+1T (α)

��

// 0

T (Jm) // T (N) // Rm+1T (B) // 0

Proof. Both statements follow easily from the naturality of the connecting morphism with respectto morphisms of exact sequences (note that the right hand diagram only makes sense for T leftexact).

8.1 Acyclic Resolutions

Definition 15. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B an additive functor. If A is an object of A then a left T -acyclic resolution of A is anexact sequence

· · · −→ F1 −→ F0 −→ A −→ 0

with all Fi left T -acyclic. Alternatively if A has enough injectives then a right T -acyclic resolutionof A is an exact sequence

0 −→ A −→ I0 −→ I1 −→ · · ·

with all Ii right T -acyclic.

The next result says that we can calculate left derived functors using acyclic resolutions. Animportant application of this result is the calculation of Tor groups from flat resolutions.

Proposition 53. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B a right exact functor. Suppose that we have a left T -acyclic resolution of an object Aof A

F : · · · −→ F1 −→ F0 −→ A −→ 0

Then there are canonical isomorphisms τn : LnT (A) −→ Hn(TF ) for n ≥ 0. These isomorphismsare natural in A, in the following sense: given a commutative diagram in A with exact rows andall Fi, Gi left T -acyclic

· · · // F1

ψ1

��

// F0

ψ0

��

// A

α

��

// 0

· · · // G1// G0

// B // 0

the following diagram commutes for n ≥ 0

LnT (A)

LnT (α)

��

+3 Hn(TF )

Hn(Tψ)

��LnT (B) +3 Hn(TG)

31

Proof. Here TF denotes the chain complex · · · −→ T (F1) −→ T (F0) −→ 0 in B. Fix an assignmentof projective resolutions P with respect to which all left derived functors are calculated. Since Tis right exact there is a canonical isomorphism τ0 : L0T (A) ∼= T (A) ∼= H0(TF ).

For n = 1 let µ : K −→ Fn−1 be the kernel of F0 −→ A, and for n ≥ 1 let µ : K −→ Fn−1 bethe kernel of Fn−1 −→ Fn−2. So for n ≥ 1 we have an exact sequence

0 −→ K −→ Fn−1 −→ · · · −→ F0 −→ A −→ 0

Let e : Fn −→ K be the unique factorisation of ∂ : Fn −→ Fn−1. Then we have a commutativediagram with exact rows

T (Fn+1)

��

// T (Fn)

T (∂)

��

T (e) // T (K) //

T (µ)

��

0

0 // 0 // T (Fn−1) +3 T (Fn−1)

It follows from the Snake Lemma that the following sequence is exact

T (Fn+1) −→ KerT (∂) −→ KerT (µ) −→ 0

and therefore we have a canonical isomorphism Hn(TF ) ∼= KerT (µ). Using the exact sequence(30) of Proposition 47 (with m = n−1) we have a canonical isomorphism KerT (µ) ∼= LnT (A), andtherefore by composition we have the required canonical isomorphism τn : LnT (A) ∼= Hn(TF ).Naturality in A follows from Proposition 48.

Proposition 54. Let A be an abelian category with enough injectives, B an abelian category andT : A −→ B a left exact functor. Suppose that we have a right T -acyclic resolution of an object Aof A

I : 0 −→ A −→ I0 −→ I1 −→ · · ·Then there are canonical isomorphisms σn : RnT (A) −→ Hn(TI) for n ≥ 0. These isomorphismsare natural in A, in the following sense: given a commutative diagram in A with exact rows andIi, J i right T -acyclic

0 // A

α

��

// I0

ψ0

��

// I1

ψ1

��

// · · ·

0 // B // J0 // J1 // · · ·the following diagram commutes for n ≥ 0

RnT (A)

RnT (α)

��

+3 Hn(TI)

Hn(Tψ)

��RnT (B) +3 Hn(TJ)

Proof. Here TI denotes the cochain complex 0 −→ T (I0) −→ T (I1) −→ · · · in B. Fix anassignment of injective resolutions I with respect to which all right derived functors are calculated.Since T is left exact there is a canonical isomorphism σ0 : R0T (A) ∼= T (A) ∼= H0(TI).

For n = 1 let µ : In−1 −→ C be the cokernel of A −→ I0, and for n ≥ 1 let µ : In−1 −→ C bethe cokernel of In−2 −→ In−1. So for n ≥ 1 we have an exact sequence

0 −→ A −→ I0 −→ · · · −→ In−1 −→ C −→ 0

Let e : C −→ In be the unique factorisation of ∂ : In−1 −→ In. Then we have a commutativediagram with exact rows

T (In−1) +3

T (µ)

��

T (In−1) //

T (∂)

��

0 //

��

0

0 // T (C)T (e)

// T (In) // T (In+1)

32

It follows from the Snake Lemma that the following sequence is exact

0 −→ CokerT (µ) −→ CokerT (∂) −→ T (In+1)

But by Lemma 7 the canonical morphism Hn(TI) −→ CokerT (∂) is also a kernel of the morphismCokerT (∂) −→ T (In+1), so we have a canonical isomorphism Hn(TI) ∼= CokerT (µ). Using theexact sequence (32) of Proposition 51 we have a canonical isomorphism CokerT (µ) ∼= RnT (A), andtherefore by composition we have the required canonical isomorphism σn : RnT (A) ∼= Hn(TI).Naturality in A follows from Proposition 52.

Remark 1. The reader may object that the definition of the isomorphism RnT (A) −→ Hn(TI)is incredibly opaque, and they’d be right. Here is the problem. Given a right T -acyclic resolution

Q : 0 −→ A −→ Q0 −→ Q1 −→ · · ·

and an injective resolutionI : 0 −→ A −→ I0 −→ I1 −→ · · ·

There is a morphism of complexes s : Q −→ I lifting the identity, which yields morphismsHnT (s) : Hn(TQ) −→ Hn(TI) = RnT (A). But it is not clear to me how to show directly thatthis is an isomorphism, which is why we resort to the construction above. It seems very likelythat σn is just HnT (s)−1 up to a sign, but checking this might require some patience (one canimmediately reduce to the case I = Q which should not be too difficult).

In most cases where we actually use acyclic resolutions to calculate, one manages somehow toavoid this issue and show HnT (s) is an isomorphism by other means. See for example Remark 2.It is possible to show that HnT (s) is always an isomorphism, but the easy proof (DTC2,Lemma44) requires some more sophisticated category theory (namely, derived categories).

9 Change of Base

We have defined the derived functors of additive functors T : A −→ B between abelian categories.Given a functor U : B −→ C (a “change of coefficients”) what is the relationship between thefunctors U ◦ LnT and Ln(UT )? In the case where U is faithful and exact we would expect thisrelationship to be an isomorphism, and that is what we shall prove in this section. First we needsome technical results.

Proposition 55. Let A be an abelian category, and suppose there is a commutative diagram withexact rows

A′α1 //

d′

��

A

d

��

α2 // A′′ //

d′′

��

0

0 // B′ β1 // Bβ2 // B′′

(33)

The connecting morphism ω : Kerd′′ −→ Cokerd′ is independent of the kernels and cokernelschosen for d′, d, d′′ in the sense that if Ker,Coker denote another set of choices, the followingdiagram commutes

Kerd′′

��

ω // Cokerd′

��Kerd′′ ω

// Cokerd′

where ω is the connecting morphism and the vertical isomorphisms are canonical.

Proof. This follows immediately from naturality of the connecting morphism with respect tomorphisms of the diagram (33). In this case the morphism is the identity, but we choose differentsets of kernels and cokernels for each copy.

33

Corollary 56. Suppose U : B −→ C is a faithful exact functor between abelian categories, andsuppose there is a commutative diagram D with exact rows in B

A′α1 //

d′

��

A

d

��

α2 // A′′ //

d′′

��

0

0 // B′ β1 // Bβ2 // B′′

(34)

Let ω : Kerd′′ −→ Cokerd′ be the canonical connecting morphism and ρ : KerU(d′′) −→CokerU(d′) the canonical connecting morphism of the diagram UD. Then the following diagramcommutes

U(Kerd′′)

��

U(ω) // U(Cokerd′)

��KerU(d′′)

ρ// CokerU(d′)

where the vertical isomorphisms are canonical.

Proof. When we refer to the “canonical” connecting morphisms we mean the morphisms con-structed using the canonical kernels and cokernels in B, C. Using the previous Proposition itsuffices to show that U(ω) is the connecting morphism τ for the diagram UD in C where the cho-sen kernels and cokernels are the images under U of the canonical ones in B. Let γ : Kerd′′ −→ A′′

and ε : B′ −→ Cokerd′ be canonical in B and let W be the walk γ, α2, d, β1, ε. Denote by UWthe walk in C given by taking the images of the morphisms in W . We use the unique propertiesof the morphisms ω, τ given in our Diagram Chasing notes.

Let A be a small, full, abelian subcategory of B containing the diagram (34) and our canonicalkernels and cokernels for d′, d, d′′ and let E be a small, full, abelian subcategory of C containing allthe objects U(A), A ∈ A. We know that W is a function walk in A and UW is a function walk inE (this is what we proved in our Diagram Chasing notes on the Snake Lemma). Let S : E −→ Abbe an exact imbedding. The composite SU : A −→ Ab is exact and faithful but not necessarilydistinct on objects, but we can find a naturally equivalent functor Q : A −→ Ab which is an exactimbedding (Mitchell II, 10.4). We have a commutative diagram of relations in Ab

SU(Kerd′′)

��

S(UW )// SU(Cokerd′)

��Q(Kerd′′)

Q(W )// Q(Cokerd′)

But Q(W ) = Q(ω) and S(UW ) = S(τ) and by naturality of the isomorphism Q ∼= SU it followsthat SU(ω) = S(τ) so U(ω) = τ , as required.

Proposition 57. Let U : B −→ C be an exact functor between abelian categories. Then for everychain complex X and n ∈ Z there is a canonical isomorphism U(Hn(X)) ∼= Hn(UX) natural inX. The following diagram of functors therefore commutes up to natural equivalence

ChB U //

Hn

��

ChCHn

��B

U// C

Proof. We choose canonical structures for B, C with respect to which all homology objects aredefined. Let X be a chain complex in B and n ∈ Z. Exactness of U means that there are

34

canonical isomorphisms U(Im∂n+1) ∼= ImU(∂n+1) and U(Ker∂n) ∼= KerU(∂n) which induce anisomorphism U(Hn(X)) −→ Hn(UX) for fitting into a commutative diagram

U(Im∂n+1)

��

// U(Ker∂n)

��

// U(Hn(X))

��ImU(∂n+1) // KerU(∂n) // Hn(UX)

These isomorphisms are easily checked to be natural in the chain complex X.

Proposition 58. Let U : B −→ C be a faithful exact functor between abelian categories. Supposewe have an exact sequence of chain complexes in B

0 −→ X ′ −→ X −→ X ′′ −→ 0

Let ωn : Hn(X ′′) −→ Hn−1(X ′) be the canonical connecting morphisms for this sequence, andρn : Hn(UX ′′) −→ Hn−1(UX ′) the canonical connecting morphisms for its image in C. Then thefollowing diagram commutes for n ∈ Z

U(Hn(X ′′))

��

U(ωn)// U(Hn−1(X ′))

��Hn(UX ′′)

ρn

// Hn−1(UX ′)

Proof. Form the diagram (8) of Theorem 26 for the sequence 0 −→ X ′ −→ X −→ X ′′ −→ 0in B and map it to C using U . Then the middle two squares give a diagram isomorphic to theanalogous diagram constucted for 0 −→ UX ′ −→ UX −→ UX ′′ −→ 0 in C (the isomorphismsbeing of the form U(Coker∂n+1) ∼= CokerU(∂n+1) and U(Ker∂n−1) ∼= KerU(∂n−1)). Thismorphism of diagrams induces morphisms of the kernels and cokernels of the vertical maps, and itis easily checked that these are the isomorphisms are the ones of the form U(Hn(Y )) −→ Hn(UY )defined above. The result now follows from naturality of the connecting morphism with respectto morphisms of diagrams of this sort, and the proof of Corollary 56 which shows that U(ωn) isin fact the connecting morphism for one of these diagrams.

Proposition 59. Let A be an abelian category with enough projectives and U : B −→ C an exactfunctor between abelian categories. Let T : A −→ B be an additive functor. Then for n ≥ 0there is a canonical natural equivalence Ln(UT ) ∼= U ◦LnT . If U is faithful and there is an exactsequence in A

0 // A′ϕ // A

ψ // A′′ // 0

Then the following diagram commutes for all n ≥ 1

· · · // U(LnTA′)

��

// U(LnTA)

��

// U(LnTA′′)

��

U(ωn) // U(Ln−1TA′) //

��

· · ·

· · · // Ln(UT )(A′) // Ln(UT )(A) // Ln(UT )(A′′)ωn

// Ln−1(UT )(A′) // · · ·

Proof. Let P be an assignment of projective resolutions with respect to which the derived func-tors are calculated. Given an object A with resolution P let ϕA : U(Hn(TP )) −→ Hn(UTP )be the canonical isomorphism defined in Proposition 57. Since U(Hn(TP )) = U(LnT (A)) andHn(UTP ) = Ln(UT )(A) the first part of the proof follows easily from the fact that this isomor-phism is natural in the chain complex.

If U is faithful then the second claim follows from the previous Proposition and the constructionof the connecting morphisms in Theorem 34.

35

Proposition 60. Let A be an abelian category with enough projectives and U : B −→ C a faithfulexact functor between abelian categories. Let T ′, T, T ′′ : A −→ B be additive functors and supposethe following sequence is exact on projectives

T ′ −→ T −→ T ′′

Then for every object A the following diagram commutes for all n ≥ 1

· · · // U(LnT ′A) //

��

U(LnTA) //

��

U(LnT ′′A)U(ωn) //

��

U(Ln−1T′A) //

��

· · ·

· · · // Ln(UT ′)(A) // Ln(UT )(A) // Ln(UT ′′)(A)ωn

// Ln−1(UT ′)(A) // · · ·

Proof. Let an object A be given, and let P be the projective resolution of A. Then the sequenceof chain complexes 0 −→ T ′P −→ TP −→ T ′′P −→ 0 is exact and the ωn are the connectingmorphisms of the corresponding long exact sequence. So the result follows immediately fromProposition 58.

There are dual results for right derived functors

Proposition 61. Let U : B −→ C be an exact functor between abelian categories. Then for everycochain complex X and n ∈ Z there is a canonical isomorphism U(Hn(X)) ∼= Hn(UX) natural inX. The following diagram of functors therefore commutes up to natural equivalence

coChB U //

Hn

��

coChC

Hn

��B

U// C

Proposition 62. Let U : B −→ C be a faithful exact functor between abelian categories. Supposewe have an exact sequence of cochain complexes in B

0 −→ X ′ −→ X −→ X ′′ −→ 0

Let ωn : Hn(X ′′) −→ Hn+1(X ′) be the canonical connecting morphisms for this sequence, andρn : Hn(UX ′′) −→ Hn+1(UX ′) the canonical connecting morphisms for its image in C. Then thefollowing diagram commutes for n ∈ Z

U(Hn(X ′′))

��

U(ωn)// U(Hn+1(X ′))

��Hn(UX ′′)

ρn// Hn+1(UX ′)

Proposition 63. Let A be an abelian category with enough injectives and U : B −→ C an exactfunctor between abelian categories. Let T : A −→ B be an additive functor. Then for n ≥ 0there is a canonical natural equivalence Rn(UT ) ∼= U ◦RnT . If U is faithful and there is an exactsequence in A

0 // A′ϕ // A

ψ // A′′ // 0

Then the following diagram commutes for all n ≥ 0

· · · // U(RnTA′)

��

// U(RnTA)

��

// U(RnTA′′)

��

U(ωn) // U(Rn+1TA′) //

��

· · ·

· · · // Rn(UT )(A′) // Rn(UT )(A) // Rn(UT )(A′′)ωn

// Rn+1(UT )(A′) // · · ·

36

Proposition 64. Let A be an abelian category with enough injectives and U : B −→ C a faithfulexact functor between abelian categories. Let T ′, T, T ′′ : A −→ B be additive functors and supposethe following sequence is exact on injectives

T ′ −→ T −→ T ′′

Then for every object A the following diagram commutes for all n ≥ 0

· · · // U(RnT ′A) //

��

U(RnTA) //

��

U(RnT ′′A)U(ωn) //

��

U(Rn+1T ′A) //

��

· · ·

· · · // Rn(UT ′)(A) // Rn(UT )(A) // Rn(UT ′′)(A)ωn

// Rn+1(UT ′)(A) // · · ·

10 Homology and Colimits

For some necessary background to this section, the reader should consult (AC,Section 2.2).

Lemma 65. If A is a cocomplete abelian category then so are ChA and coChA. A cocone is acolimit if and only if it is a colimit pointwise. If A,B are abelian categories with A cocomplete,and if F : A −→ B is an additive functor, then

(i) If F preserves coproducts, then so do ChA −→ ChB and coChA −→ coChB.

(ii) If F preserves direct limits, then so do ChA −→ ChB and coChA −→ coChB.

(iii) If F preserves all colimits, then so do ChA −→ ChB and coChA −→ coChB.

Proof. It suffices to show that both categories have arbitrary coproducts. Let I be a nonemptyindex set, and suppose we are given chain complexes Ai for i ∈ I. For each n ∈ Z take a coproductAi,n −→ Cn and let the differential Cn −→ Cn−1 be induced by the Ai,n −→ Ai,n−1. It is nothard to check that this is a coproduct of chain complexes, and a similar construction works forcochains. The other claims are easily checked.

Lemma 66. Let A be a cocomplete abelian category with exact coproducts. Then for n ∈ Z thefunctors Hn : ChA −→ A and Hn : coChA −→ A preserve coproducts.

Proof. The functors Hn,Hn are additive, so they trivially preserves initial objects. We prove

that Hn preserves nonempty coproducts, with the proof for Hn being similar. Let {Ai}i∈I bea nonempty family of chain complexes and Ai −→

⊕iAi a coproduct. Then for each n ∈

Z the morphisms Ai,n −→ (⊕

iAi)n are a coproduct in A. Since taking coproducts is exact,the coproduct of the images and kernels for the sequences Ai give images and kernels for

⊕Ai

(although probably not the canonical ones used to calculate Hn(⊕Ai)). Taking some coproduct

of the exact sequences Im(∂n+1) −→ Ker(∂n) −→ Hn(Ai) therefore provides an isomorphism⊕Hn(Ai) ∼= Hn(

⊕Ai), and it is not hard to show that Hn(Ai) −→

⊕Hn(Ai) ∼= Hn(

⊕Ai) is

Hn(Ai −→⊕Ai), which completes the proof.

Definition 16. An abelian category A is infinite if for every object A, there is an infinite numberof objects of A isomorphic to A. This avoids technical complications when defining derivedfunctors.

Proposition 67. Let A,B be cocomplete abelian categories with exact coproducts, and suppose Ais infinite and has enough projectives. If F : A −→ B preserves coproducts, then so does LnF forn ≥ 0.

Proof. The functors LnF are additive, and therefore preserve zero objects, so we can restrict ourattention to nonempty coproducts ⊕iAi. If the chosen resolution for Ai is P i : · · · −→ P i1 −→

37

P i0 −→ Ai −→ 0 then since A has exact coproducts the following sequence is a projective resolutionfor ⊕Ai

⊕P i : · · · −→ ⊕P i1 −→ ⊕P i0 −→ ⊕Ai −→ 0

and we can assume this is the chosen resolution of ⊕Ai. The functor ChA −→ ChB induced byF preserves coproducts, as does Hn : ChB −→ B. Since the resolution ⊕P i is a coproduct inChA for the resolutions P i it follows immediately that the morphisms LnF (Ai) −→ LnF (⊕Ai)are a coproduct for n ≥ 0.

Lemma 68. Let A be a cocomplete abelian category with exact direct limits. Then for n ∈ Z thefunctors Hn : ChA −→ A and Hn : coChA −→ A preserve direct limits.

Let {Ai, πij}i∈I be a direct system of modules (right or left) and let ui : Ai −→ A be anycolimit of this family. Since the canonical direct limit is a candidate for this role, we see thatui(a) = 0 iff. πij(a) = 0 for some i ≤ j and every element of A is ρi(a) for some a ∈ Ai and somei ∈ I.

Lemma 69. Let A be a category of modules. Suppose A = lim−→Ai is a direct limit. Then thereexist projective resolutions Pi of Ai forming a direct system such that P = lim−→Pi is a projectiveresolution of A.

Proof. We assume that A is RMod or ModR for some ring R. Let ui : Ai −→ A be any directlimit. For each i let Fi be the free module on the elements of Ai, and F the free module onthe elements of A, and define epimorphisms Fi −→ Ai and F −→ A in the obvious way. Letthe morphisms πij and ui induce canonical morphisms µij : Fi −→ Fj and Fi −→ F respectively.Then the {Fi, µij}i∈I are a direct system, and we claim the morphisms Fi −→ F are a colimit. Letvi : Ai −→ C be the canonical direct limit and τ : C −→ F induced by the morphisms Fi −→ F .It suffices to show that τ is an isomorphism.

It is clear that τ is surjective since every element of A is in the image of some Ai −→ A. Supposethat for some i ∈ I there are elements b1, . . . , bn ∈ Ai such that a formal sum r1 · b1 + · · ·+ rn · bnin Fi has its image in C mapped by τ to zero in F (we assume all bi 6= 0). We can partitionthe bi up into sets G1, . . . , Gr, with each element of Gi being mapped to the same element underAi −→ A, and the coefficients for the elements of Gi all adding to zero. So we can reduce to thecase where every bi is mapped to the same element of A and r1 + · · · + rn = 0. For each pair ofindices 1 ≤ i, j ≤ n there is k with the property that πik(bi) = πjk(bj). So we can find a singleindex q with π1q(b1) = · · · = πnq(bn). It is clear that r1 · b1 + · · ·+ rn · bn ∈ Fi is mapped to zeroin Fq, and hence is also zero in C, as required.

The kernels Ker(Fi −→ Ai) and Ker(F −→ A) form another direct limit, and we can repeatthis process to produce the desired projective resolutions Pi of Ai and P of A, together withmorphisms Pi −→ Pj for i ≤ j giving a direct system in ChA, and morphisms Pi −→ P whichare a colimit in ChA.

Proposition 70. Let A be a category of modules, and let B be a cocomplete abelian category withexact direct limits. If F : A −→ B preserves direct limits, then so does LnF for n ≥ 0.

Proof. We assume that A is RMod or ModR for some ring R. Let {Ai, πij}i∈I be a direct systemwith colimit ui : Ai −→ A. Find projective resolutions P of the Ai and a projective resolutionP of A so that P = lim−→Pi as above. Calculate the LnF relative to these choices of resolution.Since the functors ChA −→ ChB and Hn : ChB −→ B preserve direct limits, the morphismsLnT (Ai) −→ LnT (A) are a direct limit for all n ≥ 0, as required.

11 Cohomology and Limits

For some necessary background to this section, the reader should consult (AC,Section 2.2).

Lemma 71. If A is a complete abelian category then so are ChA and coChA. A cone is alimit if and only if it is a limit pointwise. If A,B are abelian categories with A complete, and ifF : A −→ B is an additive functor, then

38

(i) If F preserves products, then so do ChA −→ ChB and coChA −→ coChB.

(iii) If F preserves all limits, then so do ChA −→ ChB and coChA −→ coChB.

Proof. It suffices to show that both categories have arbitrary products. As before we simply takethe arbitrary pointwise products and induce morphisms making them into a (co)chain. This isa product, which shows that any product must be a product pointwise. The other claims arestraightforward to check.

Lemma 72. Let A be a complete abelian category with exact products. Then for n ∈ Z thefunctors Hn : ChA −→ A and Hn : coChA −→ A preserve products.

Proposition 73. Let A,B be complete abelian categories with exact products, and suppose A isinfinite and has enough injectives. If F : A −→ B preserves products, then so does RnF for n ≥ 0.

Proof. The functors RnF are additive, and therefore preserve zero objects, so we can restrict ourattention to nonempty products

∏iAi. If the chosen resolution for Ai is Ii : 0 −→ Ai −→ I0

i −→I1i −→ · · · then since A has exact products the following sequence is an injective resolution for∏Ai ∏

Ii : 0 −→∏

Ai −→∏

I0i −→

∏I1i −→ · · ·

and we can assume this is the chosen resolution of∏Ai. The functor coChA −→ coChB induced

by F preserves products, as does Hn : coChB −→ B. Since the resolution∏Ii is a product in

coChA for the resolutions Ii it follows immediately that the morphisms RnF (⊕Ai) −→ RnF (Ai)are a product for n ≥ 0.

12 Delta Functors

Let A be an abelian category with enough projectives. We have associated to an additive functorT : A −→ B a sequence of additive functors L0T,L1T, . . . together with connecting morphismsωn : LnT (A′′) −→ Ln−1T (A′) for any exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0. The properway to “package” the sequence of functors and the associated connecting morphisms is the conceptof a δ-functor.

Definition 17. Let A,B be abelian categories. A homological δ-functor between A and B is asequence {Tn}n≥0 of additive functors Tn : A −→ B together with an assignment of a morphism

δn : Tn(A′′) −→ Tn−1(A′)

to every n ≥ 1 and short exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 of A, satisfying thefollowing properties

1. For each such short exact sequence in A, we have a long exact sequence

· · · // Tn(A′) // Tn(A) // Tn(A′′)δn // Tn−1(A′) // · · ·

· · · // T1(A′′)δ1 // T0(A′) // T0(A) // T0(A′′) // 0

(35)

In particular the functor T0 is right exact.

2. For every commutative diagram in A with exact rows

0 // A′ //

��

A

��

// A′′ //

��

0

0 // B′ // B // B′′ // 0

(36)

39

the following diagram commutes for all n ≥ 1

· · · // Tn(A′) //

��

Tn(A)

��

// Tn(A′′)

��

δn // Tn−1(A′)

��

// · · ·

· · · // Tn(B′) // Tn(B) // Tn(B′′)δn

// Tn−1(B′) // · · ·

Definition 18. Let A,B be abelian categories. A cohomological δ-functor between A and B is asequence {Tn}n≥0 of additive functors Tn : A −→ B together with an assignment of a morphism

δn : Tn(A′′) −→ Tn+1(A′)

to every n ≥ 0 and short exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 of A, satisfying thefollowing properties

1. For each such short exact sequence in A, we have a long exact sequence

0 // T 0(A′) // T 0(A) // T 0(A′′) δ0 // T 1(A′) // · · ·

· · · // Tn(A′′) δn// Tn+1(A′) // Tn+1(A) // Tn+1(A′′) // · · ·

(37)

In particular the functor T0 is left exact.

2. For every commutative diagram in A with exact rows of the form (36) the following diagramcommutes for all n ≥ 0

· · · // Tn(A′) //

��

Tn(A)

��

// Tn(A′′)

��

δn// Tn+1(A′)

��

// · · ·

· · · // Tn(B′) // Tn(B) // Tn(B′′)δn

// Tn+1(B′) // · · ·

Definition 19. Let A,B be abelian categories. A contravariant homological δ-functor betweenA and B is a homological δ-functor between Aop and B. Similarly a contravariant cohomologicalδ-functor between A and B is a cohomological δ-functor between Aop and B.

Example 1. (i) Let A be an abelian category with enough projectives, B an abelian categoryand T : A −→ B an additive functor. For an assignment of projective resolutions P the leftderived functors {LnT}n≥0 together with the connecting morphisms ωn defined in Theorem 34are a homological δ-functor from A to B.

(ii) Let A be an abelian category with enough injectives, B an abelian category, and letT : A −→ B be an additive functor. For every assignment of injective resolutions I the rightderived functors {RnT}n≥0 together with the connecting morphisms ωn defined in Theorem 41are a cohomological δ-functor from A to B.

Definition 20. Let A,B be abelian categories. A morphism ψ : S −→ T of homological δ-functorsis a family of natural transformations {ψn : Sn −→ Tn}n≥0 with the property that for every shortexact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 of A the following diagram commutes for n ≥ 1

· · · // Sn(A′)

��

// Sn(A) //

��

Sn(A′′)δn //

��

Sn−1(A′) //

��

· · ·

· · · // Tn(A′) // Tn(A) // Tn(A′′)δn

// Tn−1(A′) // · · ·

Morphisms of homological δ-functors can be composed in the obvious way, with the identitymorphism given pointwise by the identity natural transformation. A homological δ-functor T is

40

universal if given any homological δ-functor S and natural transformation f : S0 −→ T0, thereexists a unique morphism of homological δ-functors ψ : S −→ T with ψ0 = f .

A morphism ψ : S −→ T of cohomological δ-functors is a family of natural transformations{ψn : Sn −→ Tn}n≥0 with the property that for every short exact sequence 0 −→ A′ −→ A −→A′′ −→ 0 of A the following diagram commutes for n ≥ 0

· · · // Sn(A′) //

��

Sn(A)

��

// Sn(A′′) δn//

��

Sn+1(A′)

��

// · · ·

· · · // Tn(A′) // Tn(A) // Tn(A′′)δn

// Tn+1(A′) // · · ·

Morphisms cohomological δ-functors can be composed in the obvious way. A cohomological δ-functor T is universal if given any cohomological δ-functor S and natural transformation f :T 0 −→ S0, there exists a unique morphism of cohomological δ-functors ψ : T −→ S with ψ0 = f .

Example 2. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B an additive functor. Suppose we have two assignments of projective resolutions P,Qwith associated homological δ-functors LP , LQ of left derived functors. Then the canonical naturalequivalences ψn : LPn −→ LQn form an isomorphism of homological δ-functors ψ : LP −→ LQ. Asimilar observation holds if A has enough injectives and we have two assignments of injectiveresolutions.

Definition 21. Let F : A −→ B be an additive functor between abelian categories. A universalhomological δ-functor T between A and B together with a natural equivalence u : T0

∼= F iscalled a left satellite of F and the functors Tn : A −→ B are called left satellite functors of F .Left satellites are unique up to canonical isomorphism if they exist, in the sense that if (S, v) isanother left satellite of F there is a unique isomorphism ψ : T −→ S of homological δ-functorswith vψ0 = u.

A universal cohomological δ-functor T between A and B together with a natural equivalenceT 0 ∼= F is called a right satellite of F and the functors Tn : A −→ B are called right satellitefunctors of F . Right satellites are unique up to canonical isomorphism if they exist, since if(S, v) is another right satellite of F there is a unique isomorphism ψ : T −→ S of cohomologicalδ-functors with vψ0 = u.

Definition 22. Let A,B be abelian categories. An additive functor F : A −→ B is calledeffaceable if for each object A of A there is a monomorphism u : A −→ I such that F (u) = 0. Wecall F coeffaceable if for every A there is an epimorphism u : P −→ A such that F (u) = 0.

Example 3. Suppose that A has enough projectives and let T : A −→ B be an additive functor.Then the left derived functors LnT of T are coeffaceable for n ≥ 1 since every object A admits anepimorphism P −→ A with P projective, and LnT (P ) = 0 for n ≥ 1. Similarly if A has enoughinjectives then the right derived functors RnT are effaceable for n ≥ 1.

Theorem 74. Let T be a homological δ-functor between abelian categories A,B. If the additivefunctors Tn : A −→ B are coeffaceable for n ≥ 1 then T is universal. If T is a cohomologicalδ-functor between abelian categories with Tn effaceable for n ≥ 1 then T is universal.

Corollary 75. Let A be an abelian category with enough projectives, B an abelian category andT : A −→ B a right exact functor. Then the homological δ-functor of left derived functors{LnT}n≥0 together with the canonical natural equivalence L0T ∼= T form a left satellite of T .

Proof. Choose an assignment of projective resolutions P with respect to which all left derivedfunctors are calculated and let u : L0T −→ T be the canonical natural equivalence. Then byTheorem 74 the homological δ-functor {LnT}n≥0 together with u is a left satellite of T , since thefunctors LnT are coeffaceable for n ≥ 1.

41

Corollary 76. Let A be an abelian category with enough injectives, B an abelian category andT : A −→ B a left exact functor. Then the cohomological δ-functor of right derived functors{RnT}n≥0 together with the canonical natural equivalence R0T ∼= T form a right satellite of T .

Definition 23. Let T = {Tn}n≥0 be a homological δ-functor between abelian categories A,Band let U : B −→ C be an exact functor, where C is another abelian category. The compositesUTn : A −→ C are additive functors. Given an exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 in Alet δn : Tn(A′′) −→ Tn−1(A′) be the connecting morphism and Uδn its image under U . Then thefunctors {UTn}n≥0 together with the Uδn form a homological δ-functor between A and C whichwe denote by UT . If ψ : S −→ T is a morphism of homological δ-functors between A,B thenUψ : US −→ UT is a morphism of homological δ-functors

Similarly if U : C −→ A is an exact functor then we have additive functors TnU : C −→ Band for an exact sequence 0 −→ C ′ −→ C −→ C ′′ −→ 0 in A we have the connecting morphismδn : TnU(C ′′) −→ Tn−1U(C ′) of the exact sequence 0 −→ UC ′ −→ UC −→ UC ′′ −→ 0 in A.The functors {TnU}n≥0 together with these connecting morphisms form a homological δ-functorbetween C and B which we denote by TU . If ψ : S −→ T is a morphism of homological δ-functorsbetween A,B then ψU : SU −→ TU is a morphism of homological δ-functors.

Definition 24. Let T = {Tn}n≥0 be a cohomological δ-functor between abelian categories A,Band let U : B −→ C be an exact functor, where C is another abelian category. The compositesUTn : A −→ C are additive functors. Given an exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 inA let δn : Tn(A′′) −→ Tn+1(A′) be the connecting morphism and Uδn its image under U . Thenthe functors {UTn}n≥0 together with the Uδn form a cohomological δ-functor between A and Cwhich we denote by UT . If ψ : S −→ T is a morphism of cohomological δ-functors between A,Bthen Uψ : US −→ UT is a morphism of cohomological δ-functors.

Similarly if U : C −→ A is an exact functor then we have additive functors TnU : C −→ Band for an exact sequence 0 −→ C ′ −→ C −→ C ′′ −→ 0 in A we have the connecting morphismδn : TnU(C ′′) −→ Tn+1U(C ′) of the exact sequence 0 −→ UC ′ −→ UC −→ UC ′′ −→ 0 inA. The functors {TnU}n≥0 together with these connecting morphisms form a cohomological δ-functor between C and B which we denote by TU . If ψ : S −→ T is a morphism of cohomologicalδ-functors between A,B then ψU : SU −→ TU is a morphism of cohomological δ-functors.

We can now give an alternative proof (and slight generalisation) of the results on base change.

Proposition 77. Suppose we have a commutative diagram of abelian categories and additivefunctors where A,A′ have enough injectives, U, u are exact, T, T ′ are left exact and the functor Usends injective objects of A into right T ′-acyclic objects of A′

A

U

��

T // Bu

��A′

T ′// B′

Then there are canonical natural equivalences u ◦ Rn(T ) ∼= Rn(u ◦ T ) ∼= Rn(T ′) ◦ U for n ≥ 0.Given an exact sequence

0 −→ A′ −→ A −→ A′′ −→ 0

the following diagram commutes for n ≥ 0

· · · // uRnT (A′) //

��

uRnT (A)

��

// uRnT (A′′)

��

δn// uRn+1T (A′)

��

// · · ·

· · · // Rn(uT )(A′) //

��

Rn(uT )(A)

��

// Rn(uT )(A′′) δn//

��

Rn+1(uT )(A′) //

��

· · ·

· · · // Rn(T ′)(UA′) // Rn(T ′)(UA) // Rn(T ′)(UA′′)δn

// Rn+1(T ′)(UA′) // · · ·

42

Proof. Choose assignments of injective resolutions I, I ′ for A,A′ respectively, with respect towhich all right derived functors are calculated. Let G : A −→ B′ be the composite u ◦T = T ′ ◦U .Then using Definition 24 we have three cohomological δ-functors between A and B′

{u ◦RnT}n≥0, {RnG}n≥0, {Rn(T ′) ◦ U}n≥0

The assumption that U sends injective objects of A to right T ′-acyclic objects of A′ meansthat all three of these cohomological δ-functors are universal by Theorem 74. But all threefunctors u ◦R0T,R0G,R0(T ′) ◦ U are canonically naturally equivalent to G. So by uniqueness ofright satellites there are canonical isomorphisms of cohomological δ-functors ψ : {u ◦RnT}n≥0

∼={RnG}n≥0, ϕ : {RnG}n≥0

∼= {Rn(T ′) ◦ U}n≥0 and θ : {u ◦ RnT}n≥0∼= {Rn(T ′) ◦ U}n≥0 and

moreover θ = ϕ ◦ ψ. Commutativity of the large diagram is part of the definition of a morphismof cohomological δ-functors.

Remark 2. With the notation of Proposition 77 make the further assumption that u is faithfuland fix assignments of injective resolutions I, I ′ to the objects of A,A′ respectively. Let A be anobject of A with injective resolution

0 −→ A −→ I0 −→ I1 −→ I2 −→ · · ·

Then by assumption UI is a right T -acyclic resolution of UA. Let J be the chosen injectiveresolution of UA. Then by Theorem 19 we can lift the identity to a morphism of cochain com-plexes UI −→ J and we therefore have a canonical morphism on cohomology Hn(T ′UI) −→Rn(T ′)(UA). But T ′UI = uTI and by Proposition 61 there is a canonical isomorphismHn(uTI) ∼=u(Hn(TI)) = uRnT (A). For n ≥ 0 the composite gives a canonical morphism

µnA : uRnT (A) −→ Rn(T ′)(UA)

This morphism is natural in A, so we have a natural transformation µn : u ◦RnT −→ Rn(T ′) ◦Ufor n ≥ 0. The natural transformation µ0 is the composite of the canonical natural equivalencesu ◦R0T ∼= G ∼= R0(T ′) ◦U . Therefore to show that µn is the isomorphism θn defined in the proofof Proposition 77 for n ≥ 0, we need only show that µ is a morphism of cohomological δ-functors{u ◦RnT}n≥0 −→ {Rn(T ′) ◦ U}n≥0.

Suppose we are given a short exact sequence 0 −→ A′ −→ A −→ A′′ −→ 0 in A. Letε′ : A′ −→ I ′, ε′′ : A′′ −→ I ′′ and η′ : UA′ −→ J ′, η′′ : UA′′ −→ J ′′ be the chosen injectiveresolutions and use Corollary 40 to construct injective resolutions ε : A −→ I and η : UA −→ Jfitting into short exact sequences of cochain complexes

0 −→ I ′ −→ I −→ I ′′ −→ 00 −→ J ′ −→ J −→ J ′′ −→ 0

with each morphism lifting the appropriate morphism of A or A′. By Theorem 19 we can lift theidentities to morphisms of cochain complexes g : UI ′ −→ J , f : UI −→ J, e : UI ′′ −→ J ′′ so thatwe have a diagram of cochain complexes in B′ with exact rows

0 // uTI ′

T ′(g)

��

// uTI

T ′(f)

��

// uTI ′′

T ′(e)

��

// 0

0 // T ′J ′ // T ′J // T ′J ′′ // 0

There is no reason to expect this diagram to commute, but nonetheless by (DTC,Theorem 9) and(DTC,Proposition 10) we have a commutative diagram on cohomology

· · · // Hn(uTI ′) //

��

Hn(uTI)

��

// Hn(uTI ′′)

��

δn// Hn+1(uTI ′) //

��

· · ·

· · · // Hn(T ′J ′) // Hn(T ′J) // Hn(T ′J ′′)δn

// Hn+1(T ′J ′) // · · ·

43

Examining the construction of the connecting morphisms in Theorem 41 and using Proposition62 we see that the following diagram commutes for n ≥ 0

uRnT (A′′)

µnA′′

��

u(δn) // uRn+1T (A′)

µn+1A′

��Rn(T ′)(UA′′)

δn// Rn(T ′)(UA′)

Therefore µ : {u◦RnT}n≥0 −→ {Rn(T ′)◦U}n≥0 is a morphism of cohomological δ-functors. Sinceµ0 = θ0 it follows by definition of a universal cohomological δ-functor that µ = θ. Therefore thenatural transformations µn are all natural equivalences.

References

[1] A. Grothendieck, “Sur quelques points d’algebre homologique”, Tohoku Math. J. (2) Vol. 9(1957), 119-221

[2] P. J. Hilton, U. Stammbach, “A Course in Homological Algebra”, Graduate Texts in Mathe-matics, Vol. 4, Springer-Verlag.

[3] C. A. Weibel, “An Introduction to Homological Algebra”, Cambridge Studies in AdvancedMathematics, Vol. 38, Cambridge University Press.

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